Electromechanical Model of an Induction Machine Driven Roller Conveyor

Dominique Melot de Beauregard, Burkhard Benthaus, Alexander Conradi, Stefan Kulig

Department of Electric Drives and Mechatronics, TU Dortmund University, Dortmund, Germany {dominique.beauregard, burkhard.benthaus, alexander.conradi, stefan.kulig}@tu-dortmund.de

Abstract-Customary roller conveyors as a part of intralogistic facilities still show much room for improvement concerning the power consumption and design efficiency. Especially the electric drive is oftentimes oversized. Optimisation regarding an efficient design of the motor requires in-depth information about the coupled system of the drive itself as well as the conveyer-system comprising the rollers, the drive-belt and the conveyed boxes. By applying the Lagrange formalism on the system’s energy equations, a system of differential equations is derived and dynamically adjusted with respect to the current operating state.

LIST OF SYMBOLS

btrA set of load bearing rollers fg,slideA set of load bearing sliding rollers fg,stictA set of load bearing non-sliding rollers

number of rollers covered by single box era number of electrical circuits in the rotor esa number of electrical circuits in the stator fga number of boxes transported tra total number of rollers kb index of first roller covered by box k

ic damping constant of belt section between two adjacent rollers

sc damping constant of machine shaft B,kF external force on box k N,fg,iF normal force on load bearing roller R,fg, ,k lF friction loss force between box k and load

bearing roller l R,fg,slide,iF slide friction force between roller i and box R,fg,stict,iF transmitted force between non-sliding roller

i and box R,fr,arF friction loss force between drive disc and

belt R,fr,iF friction loss force between roller i and belt

mi current of circuit m a ,b,ci three-phase stator current

transformed stator current 0r,1r,2ri ′ transformed rotor current arI moment of inertia of drive disc tr,iI moment of inertia of roller i sI moment of inertia of machine shaft

ik spring constant of belt section between two adjacent rollers

*+L L generalised Lagrangian function ,m nL inductance between circuit m and n

Lσ leakage inductance

Lμ main-magnetising inductance fg ,km mass of transported box k frm mass of one belt section

p number of pole pairs elq vector of generalised electrical velocities ,i iq q generalised coordinates and velocities

mec mec, q q vector of generalised mechanical coordinates and velocities

mR resistance in circuit m trr radius of roller

t time elT electric torque RT bearing friction torque of single roller sT drive torque rT transformation matrix rotor sT transformation matrix stator

a ,b ,cv three-phase stator voltage 0s,1s,2sv transformed stator voltage rw number of rotor turns sw number of stator turns

mv supply voltage of circuit m fg,ky coordinate of transported box k fr,iy length of belt passed under roller i

fg,ivΔ velocity difference between sliding roller i and box

α angle of contact (drive disc – belt ) η transformation ratio λ gear ratio

s,fgμ coefficient of static friction k,fgμ coefficient of kinetic friction

ξ winding factor of first harmonic arφ angle of drive disc tr ,φ i angle of roller i sφ angle of machine shaft

I. INTRODUCTION

Roller conveyors are commonly used for transportation of packaged goods like pallets and boxes within intralogistic facilities with a maximum speed of 2m/s. The rated power of the drives is within a range of 120 to 750W [7].

Because of the vague estimates used while planning the facilities, there is a high potential for optimisation regarding the design effort and energy efficiency [3].

Here the selection of the right motor to use has a huge potential of optimisation. Up to now the maximum weight of

btra

0s,1s,2si

1071978-1-4673-1653-8/12/$31.00 '2012 IEEE

the conveyed goods is the main criterion for the choice and leads to an oversizing. This results in an operating point far away from the rated conditions and a corresponding decrease in efficiency. Usually the entire intralogistic facility consists of many conveyor tracks and for that of many drives. According to this the possible savings are huge.

Anyway the optimisation process has to fulfil the requirements of reliability and high availability of the system. Therefore the careful consideration of the whole electromechanical system is the prerequisite of a demand compliant design. Thus the model of a roller conveyor as described in the following section will be defined, where in addition to the used motor model the description of the mechanical parts of the conveyor track will be discussed.

The work and results presented here are a part of the research work of the subproject B2 of the Collaborative Research Center 696 “Logistics on Demand”. It is founded by the German Research Foundation (DFG).

II. REGARDED SETUP

The experimental setup used in this research work consists of two straight parts connected by two arcs with a total length of 13.85m. One straight part (Fig. 1: green), one arc (Fig. 1: blue) and one composed part (Fig. 1: red) are powered separately.

The rollers are tangentially driven by a textile flat belt, with little rollers beneath to get the necessary contact force. The 550W induction motor is connected by a gear box with a gear ratio of λ 17= to the drive disc which transmits the torque to the belt. A prestressing of constant 500N is provided by a device behind the drive disc to the belt.

The described modelling applies to the green coloured straight part of the experimental setup in Fig. 1. This part consists of 40 rollers and has a total length of 3m.

III. SYSTEM MODEL

The basic model of the roller conveyor is going to be derived by applying Lagrange’s formalism. It comprises a general model of the electric machine as well as the mechanical coupling between the constituent parts of the conveyor track. Hereby it is possible to adapt the model for various types of electric drives and different levels of detail concerning the description of the mechanical construction of the conveyor track.

A. General Model

1. The Euler-Lagrange Method Within the formalism of Euler-Lagrange, all electrical and

mechanical degrees of freedom of a physical system are described by generalised coordinates qi and their derivatives, the generalised velocities .iq The interdependence of the single degrees of freedom with each other is covered within the so called kinetic potential *+L L , where L denotes the conservative energy in the system at time t and *L denotes the energy exchanged between the system and its environment and up to the time t (e.g. ohmic or friction losses, energy supply).

Differentiation of the kinetic potential according to (1)

yields a system of differential equations, whose solution describes the transient behaviour of the system’s degrees of freedom [6].

2. Definition of the Generalised Coordinates A general electric machine is modelled by the mechanical

torsion angle of the rotor sφ , the aes electric circuits of the stator and the aer electric circuits of the rotor. Given a supply voltage a,b,c ,v the currents

es er1 , ..., a ai i + therefore form the machine’s electric degrees of freedom. Within the Euler-Lagrange formalism, electric currents are described as generalised velocities, such that:

(2) According to Fig. 2, the conveyor track is modelled by the

the torsion angle of the track rollers and the drive roller ( )trar tr ,1 tr ,φ ,φ , ,φ… a as well as the positions ( )trar tr ,1 tr ,, , , ay yy … of the sections of the drive belt.

( ) ( )* *

0i i

L L L Lddt q q

⎛ ⎞∂ + ∂ +⎜ ⎟ − =⎜ ⎟∂ ∂⎝ ⎠

( )es er

Tl 1e ,.. ., a ai i +=q

Fig. 2. Schematic of the conveyor track.

Fig. 1. Experimental setup of the roller conveyor.

1072

yfr,i here denotes the length of the drive belt already passed by roller i at time t. Hereby the vectors of the generalised coordinates and generalised velocities for the degrees of freedom of the conveyor track form as:

(3)

The form-fitting connection of the machine’s rotor shaft and the drive disc by a gear transmission allows the joint treatment of the angles of the machine rotor sφ and the drive disc arφ with a transmission factor λ , such that:

rs aϕ = λ ⋅ϕ (4)

3. Kinetic Potential of the System By applying the generalised coordinates defined above, the

kinetic potential of the general machine can be written as:

(5)

And for the mechanical system as:

(6)

with the proviso that:

(7)

4. The system of differential equations Applying the Euler-Lagrange operator (1) on (5) and (6)

yields the following system of differential equations comprising:

Voltage equations of the es er+a a electric circuits:

(8) Mechanical equations of the electric machine:

(9)

Mechanical equation of the drive disc:

(10)

Mechanical equations of non-load bearing rollers:

tr , tr , R , tr R ,fr ,· 0i i i iI T r Fϕ + + = (11)

Mechanical equations of load bearing rollers:

tr , tr , R , tr R ,fr , tr R ,fg , , 1· 0ki i i i k i bI T r F r F − +ϕ + + + = (12)

Segments of the drive belt:

(13)

Segment 1of the drive belt: (14)

Segment atr+1 of the drive belt:

(15)

Conveyed boxes: (16) With the exception of (9), Equations (8) - (16) can be

written in a hypermatrix structure, such that: (17) (18)

es er es er

es er

* 2s s , s

1 1kinetic energy magnetic energy

2 2s s

10 0

mechanical losses electrical losses

s s0

1 1· · · ( )· ·2 2

1 1· · d · d2 2

( )· d

+ +

= =

+

=

+ = ϕ + ϕ

+ ϕ +

− ϕ

∑ ∑

∑∫ ∫

∫

a a a a

m n m nm n

t ta a

m mm

t

c

L L I L i i

t R i t

T t tes er

1 0

driving torque energy electrical supply energy

( )· d+

=− ∑ ∫

ta a

m mm

tv t i

fgtr tr

tr tr

1* 2 2 2 2

ar ar tr , fg, fg, fr, fr,1 1 1

kinetic energy

1 12 2

1 1 0

potential energy (belt) damping

1 · · · ·2

1 1· · d2 2

aa a

i i k k j ji k j

ta a

j jj j

j j

L L I m y m yI

tck

+

= = =

+ +

= =

⎛ ⎞+ = ϕ + ϕ + +⎜ ⎟⎜ ⎟

⎝ ⎠

− Δ + Δ

∑ ∑ ∑

∑ ∑ ∫

( )

tr

fg btr

R tr,1 0

losses (belt) friction losses (bearings)

R ,fg, , tr tr, 1 fg,1 1 0

friction losses (rollers - conveyed boxes)

R ,fr, f

· d

· · d

·

k

ta

ii

a ta

k l b l kk l

i

T t

F r y t

F y

=

+ −= =

+ ϕ

+ ϕ −

+

∑∫

∑∑∫

( )

( )

tr

fg

r , tr tr,1 0

friction losses (rollers - belt)

R,fr,ar fr,ar ar,0

friction losses (drive disc - belt)

B, fg,1 0

external forces energy

ar

· d

· · d

· d

ta

i ii

t

i

a t

k kk

r t

F y r t

F y t

=

=

− ϕ

+ − ϕ

− −

∑∫

∫

∑∫ s ar0

driving torque energy

· dt

T t− λ ⋅ ϕ∫

tr

1 1tr

Tar fr,1 fr, 1 fr, fr, ar, , , ,( )

+

−

Δ Δ Δ

Δ = − … − … −j a

j j ay y y yy y

( ) [ ]es er

, es er1

· · ( ) 0 1,.,+

=

+ − = ∈ +∑a a

m m m n n mn

R i L i v t m a a

es er es er

s ss s

,s

1 1 s

· · ...

1 ... · ( ) 02

+ +

= =

ϕ ϕ⎛ ⎞ + −⎜ ⎟⎝ ⎠

− − =ϕ

∂∂∑ ∑

a a a am n

m nm n

d dd I cdt dt dt

Li i T t

( )( )

fr, fr, R,fr, fr, 1 1 fr, 1 fr, 1

fr , 1 1 fr, 1 fr, 1

· · · ·

· · · 0− + + +

− + + +

− − + + −

− + + − =i i i i i i i i i i

i i i i i i i

m y F c y c y c y

k y k k y k y

c

btr

fg, fg, R ,fg, , B,1

· 0=

− + =∑a

k k k l kl

m y F F

ss srs s s s

rs rrr r r r

v i iR L

ddt

⎧ ⎫⎪ ⎪⎡ ⎤⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎪ ⎪= + ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥

⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪⎪ ⎪⎩ ⎭

L LR 0 i iL Lv 0 R i

vi

fr mec mec

fg

mec mec

· ·

·

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤⎢ ⎥+ =⎢ ⎥⎢ ⎥⎣ ⎦

+

I 0 0 0 0 00 M 0 q 0 C 0 q0 0 M 0 0 0

0 0 00 K 0 q Q 00 0 0

s ar R,fr,ar ar ar·λ ⋅ + ⋅ = ϕT r F I

( )( )

fr,1 fr,1 R,fr,1 ar fr,1 fr,21 1 2 2

1 ar fr,1 f1 2 ,22 r

· · · ·

· · · 0

− − + + −

− + + − =

m y F c y c y c y

k y k k y k y

c

tr2,...,=i a

( )( )

fr,atr R,fr,atr+1 atr fr,atr 1 atr atr 1 fr,atr

atr 1 ar atr fr,atr 1 atr atr 1 fr,atr atr 1 ar

ar· · ·

· · · · 0

m y F c y c y

c y k y k k k y

c

y− +

+ − + +

− − + +

− − + + − =

( )( )

tr tr fg

tr tr fg

Tmec ar tr,1 tr, ar fr,1 fr, fg,1 fg,

Tmec ar tr,1 tr, ar fr,1 fr, fg,1 fg,

φ ,φ , ,φ , , , , , , ,

φ ,φ , ,φ , , , , , , ,

a a a

a a a

y y y y y

y y y y y

= … … …

= … … …q

q

1 0 7 3

(19)

Due to the products of rotor- and stator currents in the mechanical equation of the electric machine, (19) cannot be included in the matrix description, but still has to be written separately.

The system components in the hypermatrix form (18) are the matrix of the mass moments of inertia,

(20)

the matrices of the masses of the drive belt Mfr and the conveyed boxes Mfg,

(21)

The matrix of damping constants,

(22)

the matrix of spring constants

(23)

and the vector of external (non-conservative) forces of the mechanical subsystem of the conveyor track:

(24)

B. Modelling of the Induction Machine Starting with the equations of the general machine in (17) ,

the three-phase induction machine with es 3a = stator circuits and a short circuit rotor with era phases or rotor-bars can be modelled as:

(25) The dense matrices ss sr rs rr, , and L L L L represent the

magnetic flux coupling of the machine circuits with each other. The mutual inductances between the stator and rotor circuits in sr rsund L L are furthermore a function of the rotor angle sφ , thus causing non-linearity in the voltage equations (25).

There are numerous methods of transformation in order to decouple the electric phases by converting the inductance matrices into a diagonal form [2, 5], thus making it a lot easier to solve the system equations. The method used in the present paper is presented in detail in [6].

Similarly to the transformation to symmetric components, the real-valued electric system variables will be transformed into complex components of zero, positive and negative sequence:

(26)

(25) can therefore be written as:

(27)

es er es er

el

s ss s

,s

1 1 s

electric Torque

· ·

1 · ( )2

a a a am n

m nm n

T

d dd I cdt dt dt

Li i T t

+ +

= =

ϕ ϕ⎛ ⎞ +⎜ ⎟⎝ ⎠

∂+

ϕ∂= ∑ ∑

ar

tr,1

tr,

0 00

00 0 n

II

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

I

tr fg

fr,1 fg,1

fr fg

fr, 1 fg,

0 0 0 00 0 , 0 00 0 0 0+

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

=

a a

m m

m mM M

er

ss srs s ss

rs rrr r r

T Ts rc 3a 4b( , , ) ( , , )a

dd

i i i i i

t

+

⎧ ⎫⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎪ ⎪= + ⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎢ ⎥⎪ ⎪⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭

= = …

L LR 0 i ivL L0 R i i0

i i

( ) ( )

s coupls sss*coupl rr rr

T Ts r 0s 1s 2s 0r 1r 2r, , , ,i i i i i

t

i

dd⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ = + ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

= =

L LR 0 i iL L0 R i i0

i i

v

s rs r r s s s r r r s , , ,≡ ≡ ≡ ≡ =i T i i T i v TvT v v 0

( )( )

( )

tr tr

tr

tr+1 tr tr tr

1 1 1

1 2

1

2

1

1 2

0 0

00 0

0 00

0 0

+ +

+

⎛ ⎞⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

+

⎜ ⎟

− −

− +−

−

− − +⎝ ⎠

a a

a

a a a a

c

c

c c c

c c

c

c c

cc

c c

C

( )( )

( )

tr tr

tr

tr+1 tr tr tr

1 1 1

1 2

1

2

1

1 2

0 0

00 0

0 00

0 0

+ +

+

⎛ ⎞⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

+

⎜ ⎟

− −

− +−

−

− − +⎝ ⎠

a a

a

a a a a

k

k

k k k

k k

k

k k

kk

k k

K

1074

Written out in full, (27) yields the stator

(28)

and rotor equations

(29)

with (30)

and (31) In the case of a symmetric three-phase harmonic voltage

source the zero components of the voltage and the current will be zero. Moreover, the complex currents 1si and 2si as well as 1r′i und 2r′i are conjugate-complex to each other. Therefore, (28) und (29) can be equally reduced to the system of positive components:

(32)

The equations in (32) can subsequently be illustrated in the form of a T-equivalent circuit with the well-known parameters but transformed electric voltages and currents, where ( )sj φ

,r 1r1,ro t 3φ 2

pv j L i Lp e iσ μ μ+ +′= ′ represents the

additional rotationally induced voltages.

In contrast to the T-equivalent circuit with stationary phasors, the system of differential equations in (32) can be used to calculate dynamic transients.

Within the transformed system, the electric torque elT from (19) can be calculated as

(33)

C. Adjustment of the Mechanical System

1. General Assumptions The mechanical sub-system consisting of the rotor shaft,

the gear box, the drive disc and the part of the belt wound around the drive disc can be described by one generalised coordinate, where the gear box is taken into account by using the gear ratio λ . The maximum force maxF transmitted via the connection of the drive disc and the belt dependent on the prestressing force prestr ,F the angle of contact α and the coefficient of static friction sμ is given by Euler-Eytelwein’s equation as:

(34)

With the information of the experimental set-up used a maximum transmittable force of max 2kN=F or with the radius ar 0.075m=r of the drive disc a maximum torque of

max 160Nm=T is derived, which neither in the simulations nor measurement was reached. For this it can be assumed that the connection is fixed and the generalised coordinate can be defined as follows:

ar ar ars

ar arφφ λq y r r= = ⋅ = ⋅ (35)

In the following only the case of one single box being conveyed on the regarded conveyor track is discussed.

2. Neglecting Slip Between Rollers and Belt Based on the assumption that there is no slip between belt

and rollers, the following holonomic constraint can be defined for each roller:

s

s

s

0s s 0s

1s s 1s

2s s 2s

,s 0s

,s 1s

2s

,s

er j φ,max

j

0 00 00 0

0 030 02

30 02

0 0 03

0 02

0 0

pmn

p

v R iv R iv R i

L iL L i

iL L

ddt

d aL e

de

t

σ

σ μ

σ μ

+

−

⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥=⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦ ⎝ ⎠

⎡ ⎤⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥+ + ⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎝ ⎠

⎢ ⎥+⎣ ⎦

+

R

L

coupl

s

0r

1rφ

2r

iii

⎡ ⎤ ⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

L

r

r

s

r 0r

r 1r

r 2r

,r 0r

er,r 1r

2rer

,r

er j φcoupl,max

0 0 00 0 00 0 0

0 0

0 02

0 02

0 0 03

0 02

0 0

p

R iR i

R i

Li

aL L i

iaL L

aL e

ddt

ddt

σ

σ μ

σ μ

−

⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥=⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦ ⎝ ⎠

⎡ ⎤⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥′+ + ⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎝ ⎠

⎢ ⎥′+⎢ ⎥⎣ ⎦

+

R

L

* co p

s

u l

0s

1sj φ

2sp

ii

e i+

⎡ ⎤ ⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

L

22r r

s sμ μ μ

⎛ ⎞ξ′ = = η⎜ ⎟ξ⎝ ⎠

wL L L

w

r rcoupl,max

s s

.w

L L Lw μ μ

⎛ ⎞ξ= = η⎜ ⎟ξ⎝ ⎠

( )s sj φ j φ* *el 1s 1r 2s 2r

32

p pT jpL i i e i i e−μ ′ ′= −

s

s

,ss 1s 1s 1s

r 1r 1r,r

j φ 1s

j φ 1r

3 00 d200 3 d0

2

03 d2 d 0

p

p

L LR i iR i itL L

v

ieL

it e

σ μ

σ μ

μ

+

−

⎡ ⎤+⎢ ⎥⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥′ ′ ′⎝ ⎠ ⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠′ +⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤ ⎛ ⎞⎪ ⎪+ ⎨ ⎬⎢ ⎥ ⎜ ⎟′⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

sμmax prest

(r

α)( 1)FF e ⋅⋅= −

Fig. 3. Dynamic T-equivalent circuit of the induction machine.

1075

(36)

Thus the number of degrees of freedom can be reduced by leaving out the coordinates and velocities of the belt, leading to the final vector of mechanical generalised coordinates:

(37)

The system of equations in (18) simplifies to:

(38)

3. Coupling of Rollers and Conveyed Box The coupling of the box and the load bearing rollers is

either force-locked or can be described taking the dynamic

friction into account, i.e. the rolling friction is neglected consciously (losses can be taken into account within the bearing friction of the rollers), as depicted in Fig. 4.

Therefore the force R,fg,iF between the rollers and the box described in the general mechanical model corresponds to an interior force of the system, which is not calculated separately, but will be taken into account by establishing a mechanical sub-system consisting of the load bearing rollers and the box connected by the holonomic constraints:

(39)

The coupling of the sliding rollers is carried out with respect to the friction force R ,fg ,slide,iF whose direction depends on the velocity difference fg,Δ jv between the roller and the box:

fg , tr , tr fg btrφ ·i iv r y i AΔ = − ∈ (40)

With the normal force N,fg,iF acting on the roller and the dynamic friction coefficient k,fgμ the friction force is calculated as follows:

(41)

4. Identifying the Contact Status of the Rollers The following procedure is used each time step to

determine the sets of sticking fg,stictA and sliding fg,slideA rollers, taking the contact status of the previous time step into account.

Fgdkjgfkd sdfsfsdfsfssdfds

The first step is to evaluate the change in the set of load bearing rollers btrA caused by the movement of the box.

(42)

In the second step it has to be checked if the rollers which

were sliding in the previous time step are now sticking. Therefore the velocity difference fg,Δ iv between roller and box is evaluated in (40).

If the comparison of the actual and the previous time step shows a change of sign in the velocity differences fg,ivΔ , the corresponding roller is moved into the set of sticking rollers

fg,stictA . In case of an existing velocity difference fg , 0Δ ≠iv in the current time step acceleration is applied to align the velocities. The resulting sets are:

(43)

The next step is to identify which of the sticking rollers has to be reclassified as sliding. To this end the spring- and damper forces of the belt are determined

(44)

and used to calculate the acceleration for the subsystem consisting of the box and all sticking rollers (taking the forces of the sliding rollers into account).

(45)

Now the forces transmitted from the sticking rollers to the box R ,fg ,stict ,iF are determined:

(46)

These forces are then compared to the maximum transmittable Forces R ,fg,stict , ,maxiF . All rollers which violate the constraint

(47)

are moved from the set fg,stictA to set fg,slideA , resulting in the following distribution:

(48)

[ ]{ }{ }

fg,slide,0 fg,slide btr btr btr

fg,stict,0 fg,stict btr

: ( 1) ( ) ( ) \ ( 1)

: ( 1) ( )

⎡ ⎤= − ∩ ∪ −⎣ ⎦

= − ∩

A A t A t A t A t

A A t A t

( ) ( ){ }( )

fg,slide,1 fg,slide,0 fg, fg,

fg,stict ,1 fg,stict ,0 fg,slide,0 fg,slide,1

: | sgn ( 1) sgn ( )

: \

⎡ ⎤= ∈ ∧ Δ − = Δ⎣ ⎦

= ∪

i iA i i A v t v t

A A A A

( )( )

, tr tr, 1 1 tr, 1 tr, 1tr

k, tr tr, 1 1 tr, 1 tr, 1

· φ φ φ

· φ φ φc i i i i i i i i

i i i i i i i i

F r c c c ci A

F r k k k k− + + +

− + + +

⎡ ⎤= − + + −⎣ ⎦ ∈⎡ ⎤= − + + −⎣ ⎦

fg ,slide,1 fg ,stict ,1

fg ,stict ,1

R ,fg,slide, B R, , k ,tr

fg,1

fg tr,2tr

1

1

i i c i ii A i A

ii A

F F T F Fr

ym I

r

∈ ∈

∈

⎛ ⎞− − + +⎜ ⎟

⎝ ⎠=+

∑ ∑

∑

tr, R ,R,fg,stict , fg,1 , k , fg,stict ,12

trtr

i ii c i i

I TF y F F i A

rr= − − − − ∈

R,fg,stict , ,max N,fg s,fg, μi iF F< ⋅

( ){ }( )

fg,slide fg,slide,1 fg,stict ,1 R,fg,stict , R ,fg,stict ,max

fg,stict fg,stict ,1 fg,slide,1 fg,slide

( ) : ( ) | ( ) ( )

( ) : ( ) \ ( ) ( )

= ∪ ∈ ∧ >

= ∩

iA t A t i i A t F t F

A t A t A t A t

fr, tr tr,φ= ⋅i iy r

( )tr fg

Tmec,red ar tr,1 fr, 1 fg,1 fg,φ ,φ , , , , , += … …a ay y yq

mec,red mec,red

mec,red mec,red

· ·

·

⎡ ⎤ ⎡ ⎤+⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤+ + =⎢ ⎥⎣ ⎦

fg

0 0q

0 M 0 0I

K 0Q

C

q 00 0

q

fg tr tr, fg tr tr,·φ , ·φn ny r y r==

( )R,fg,slide, fg, N,fg, k,fg fg,slide sgn · ·μi i iF v F i A= Δ ∈

Fig. 4. Schematic on the connection of rollers and box.

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This finally results in the acceleration of the box and coupled rollers

(49)

and the angular acceleration of the slipping rollers.

(50)

D. Solving the System of Differential Equations The parameters of the electric drive were determined using

data supplied by the manufacturer and measurements as well as computation done by the authors. Most parameters of the mechanical System were determined using measurements. The parameters of the belt were determined as part of the subproject B4 of the Collaborative Research Center 696 “Logistics on Demand” [1]. Not all parameters could be measured. The missing parameters had to be obtained by using default values [4] which had to be adjusted by comparing simulated and measured data.

The solution of the system of differential equations is computed using a solver for ordinary differential equations (ode) in Matlab. In this case the ode-45-solver, which is based on the Runge-Kutta method, was chosen, because it offers a good accuracy coupled with a high stability of the solving process.

The analysis of the simulation results and the measurements was also done in Matlab, which allows for an efficient comparison of the results.

IV. COMPARISON TO MEASUREMENTS A. Measurement Setup

As well as the simulation, the measurement setup is limited to the part of the experimental setup described in section II. Regarding mainly the behaviour of the electrical machine, the first setup only considers the relevant data for this.

The electrical input quantities, thus the three phase voltages and currents, as well as the mechanical output quantities rotational speed and torque between gear box and drive disc were detected.

B. Measurements Especially the observation of dynamic transients will allow

for an analysis of the coupled system consisting of drive and conveyor track. Therefore a transient startup from standstill of the system has been performed and measurements have been taken. Each test has been carried out with a single box with a mass of 26 kg and 52 kg respectively on the conveyor track.

The box is placed at the starting end of the considered straight track segment. For startup, the feeding voltage is directly applied to the drive, without the use of power electronics. The total time of traverse over the considered track segment is approximately 4 seconds, wherein the steady-state is reached in less than one second.

C. Results In Fig. 6, the measurements of the test conveyor show a

good accordance with the simulation results. The deviations in the amplitudes can be explained by the limited knowledge of the parameters of the mechanical components and the not considered slip between the rollers and the drive belt at this state of the model.

The latter could in particular explain the higher torque towards the end of the acceleration transient in the measurement for the 52 kg box (Fig. 6), which does not occur in the simulation results

A higher mass of the conveyed box results in increased stiction between the box and the rollers, whereas the maximum transmittable force between the belt and the rollers stays the same. However, an increase in the mass of the box requires higher acceleration forces transmitted between belt and rollers, thus making the occurrence of slip more likely.

This in turn would cause increased friction losses within the system, which would lead to a prolonged stage of

( )( ) })

fg ,slide fg ,stict

fg ,stict

fg R,fg,slide, B R,tr

fg tr,2tr

tr tr , 1 1 tr , 1 tr, 1

tr tr , 1 1 tr , 1 tr , 1

1 11

φ φ φ

φ φ φ

j ij A i A

ii A

i i i i i i i

i i i i i i i

y F F Trm

r

r c c c c

k

I

r k k k

∈ ∈

∈

− + + +

− + + +

⎛= − −⎜⎜

⎝+

⎡ ⎤+ − + + −⎣ ⎦

⎡ ⎤+ − + +

⎧

−⎣ ⎦

⎨⎩

∑ ∑∑

(

( )( ) )

tr, tr R ,fg,slide, R ,tr,

2tr tr , 1 1 tr, 1 tr, 1

2tr tr, 1 1 tr , 1 tr, 1

1φ ·

φ φ φ

φ φ φ

i i ii

j i j i i i i

i i i ii ii

r F T

r c c c c

r k k k k

I

− + + +

− + + +

= − +

⎡ ⎤+ − + + −⎣ ⎦

⎡ ⎤+ − + + −⎣ ⎦

0 0.2 0.4 0.6 0.8 1

-15

0

15

30

45

60

Time t [s]

Tor

que

T s [

Nm

]

Simulation

Measurement

0 0.2 0.4 0.6 0.8 1-30

-15

0

15

30

45

60

Time t [s]

Tor

que

T s [

Nm

]

Simulation

Measurement

Fig. 6. Results of the simulation and measurements of the transient startupwith fg 52 kg.m =

Fig. 5. Results of the simulation and measurements of the transient startup with fg 26 kg.m =

1077

acceleration, as parts of the torque during the transient are used for balancing the friction forces.

Fig. 7 shows the angular velocity of the rollers and the drive disc (reduced by factor tr arr r ) during the acceleration transient. Expectedly, the rollers accelerate successively, whereat, caused by the isotropic spring stiffness, the first and last as well as all other complementary roller pairs are each accelerated at similar instants of time and with similar magnitude of acceleration. In further research on the topic it will be analyzed to which degree the observed behaviour will change under consideration of anisotropic spring stiffness and applied prestressing of the belt.

V. SUMMARY AND PERSPECTIVES

A method for the description of the electromechanical behaviour of roller conveyors carrying different loads was presented. Based on the energy equations of the system, the differential equations were set up taking the slip between the rollers and the box into account.

The comparison of the simulation results with measurements showed good agreement, thus proving the model to be adequate for the computation of transients.

In future investigations it has to be checked whether taking the slip between the rollers and the belt into account or using non-isotropic spring constants in conjunction with a prestressing of the belt will lead to better results. Though keeping in mind that this will cause an increase in the number of parameters of the system.

REFERENCES [1] G. Fischer, R. Zielke, H.-G. Rademacher, “Zustandsüberwachung von

Komponenten intralogistischer Systeme“, Belastungsabhängige Auslegung, Überwachung und Steuerung von intralogistischen Systemen, Dortmund : Verl. Praxiswissen, 2008.

[2] K.P. Kovacs, I. Racz, Transiente Vorgänge in Wechselstrommaschinen, Verlag der Ungarischen Akademie der Wissenschaften, Budapest 1959.

[3] B. Ku�nne, D. Wieczorek, "Research to optimize the embodiment design of modules and components used in roller conveyors", Automation and Logistics (ICAL), 2010 IEEE International Conference on , vol., no., pp.495-500, 16-20 Aug. 2010.

[4] D. Muhs, H. Wittel, D. Jannasch, J. Voßiek, Roloff/Matek Maschinenelemente. 19. Auflage. Vieweg, Wiesbaden 2009.

[5] P. H. Park, “Two reaction theory of synchronous machine“, AIEE Trans. Vol. 48, PT. 1, PP. 716-730, Jul. 1929.

[6] A. Puchala, Dynamics of machines and electromechanical systems, PWN Warsaw, 1977.

[7] M. ten Hompel, T. Schmidt, L. Nagel, R. Jünemann,Materialflusssysteme – Förder- und Lagertechnik, 3., völlig neu bearb. Aufl., Berlin [u.a.]: Springer, 2007.

0 0.05 0.1 0.15 0.20

10

20

30

40

50

Time t [s]

Rot

atio

nal

Spe

ed ω

[r

ad/s

]

Rol

ler

Inde

x i

5

10

15

20

25

30

35

40

Fig. 7. Rotational speed of the rollers and the drive disc.

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