Barring Inching Drive

29
MILL SELF-DISENGAGEMENT BARRING SYSTEMS JOHANN J. van RENSBURG ENGINEERING MANAGER MET. PLANTS ANGLOGOLDASHANTI CONTINENTAL AFRICA REGION Jan 2013

description

Barring Inching Drive

Transcript of Barring Inching Drive

  • MILL SELF-DISENGAGEMENT

    BARRING SYSTEMS

    JOHANN J. van RENSBURG

    ENGINEERING MANAGER MET. PLANTS

    ANGLOGOLDASHANTI

    CONTINENTAL AFRICA REGION

    Jan 2013

  • CONTENT

    1. ABSTRACT

    2. INTRODUCTION

    3. BACKGROUND

    4. MILL DETAIL

    5. PRINCIPLE and OPERATION OF SELF-DISENGAGEMENT BARRING SYSTEMS

    6. METHODOLOGY

    7. COST MODEL

    8. ADVANTAGES OF SELF-DISENGAGEMENT BARRING SYSTEMS

    9. DIS-ADVANTAGES OFSELF-DISENGAGEMENT BARRING SYSTEMS

    10. INSTRUMENTATION and MEASUREMENTS

    11. MEASUREMENT RESULTS

    12. RESULTS

    13. CONCLUSION

    14. FINAL WORD

    15. ACKNOWLEDGEMENT

  • 1.ABSTRACT

    Grinding Mills are production critical machines in not just Gold mining Process

    Plants but in every plant where they are utilised in order to meet business

    objectives. They in a sense are the heart of plants and in order to achieve

    business objectives are required to be available well above 90 %.

    These machines have been in operation for close to a century and in this time

    with a diameter scale up of at least five fold. A typical or favoured drive

    arrangement feature a wound rotor induction motor driving a girth gear via a

    single or dual-stage gearbox. This drive train configuration in itself is about

    forty years in use.

    However numerous improvements and changes occurred during passed years

    at various operations on mills from various manufacturers. Large diameter

    Wrap-around motor designs with huge diameters are just one of these

    interventions. These designs eliminate the use of gearboxes, pinions and girth

    gears.

    Evolution of Mill technology

    Conventional Milling First generation SAG Milling

    Latest Generation SAG Milling

  • 14,7Meter diameter wrap-around motor mill design.

    Latest development in drives is sophisticated direct motor drives onto the girth

    gear without the use of Gearboxes and pinions.

    medium voltage frequency converter with exclusive DTC (direct torque

    control).Direct motor drives with frequency torque control.

    around motor mill design.

    Latest development in drives is sophisticated direct motor drives onto the girth

    gear without the use of Gearboxes and pinions. This latest generation is a

    medium voltage frequency converter with exclusive DTC (direct torque

    control).Direct motor drives with frequency torque control.

    Latest development in drives is sophisticated direct motor drives onto the girth

    test generation is a

    medium voltage frequency converter with exclusive DTC (direct torque

  • This unfortunately does not exclude us improving and maintaining current

    plants with old designs and operating principles. It is appropriate to constantly

    apply comprehensive design and manufacturing quality assurance to achieve

    reliable mill drives. Further to this plant operating practices mill reliability

    should be taken into account.

    This paper summarises the investigation into some of these old features in

    design and operating principles. The findings indicate that some of our

    practices over years have negatively impacted on our reliability without almost

    realising it. Almost too small an issue to pay attention to.

    Another important factor to be considered is that these machines are all

    purpose-built and as such in many cases prototypes. It is necessary to

    undertake independent design audits and apply rigorous quality assurance and

    to assess operating practises.

    This abstract have focused on drives and their improvements as a result of

    operating practise audits that were conducted. One of the components of

    current drive trains is Barring and the paper summarises the investigation into

    Barring specifically and the effect thereof on the reliability and sustainability of

    Gearbox life.

  • 2.INTRODUCTION

    Focussing on the demands of grinding mills we can classify them into the

    following.

    Operating demands

    Maintenance demands

    Protective demands

    As per the abstract this paper will focus on the requirements of Operating

    demands. One of the critical tasks from an Operational demand identified in

    the start-up procedure of grinding mills is Barring.

    The Barring drive, also known as Inching drive, Sunday-drive or auxiliary-

    drive, is an important component of any mill installation. It is used for

    maintenance and inspection purposes, as well as an emergency auxiliary drive

    to keep the mill rotating when the main motor fails and it is required for the

    mill to rotate at certain intervals. Re-lining as a maintenance function cannot

    be done without a barring drive.

    Probably the most important function of the Barring drive is to dislodge Frozen

    or Lock charge thus as frozen or lock charge protection.

    Locked or Frozen charge is a common occurrence with mill applications and is

    therefore it is an advantage that the Barring drive can offer protection

    againstthis condition. Frozen charge can occur when the mill has beenstopped

    long enough for the product to solidify. If the mill is startedagain this solidified

    product may fall and damage the mill lining. This is achieved by the several

    slow rotations (1turn per minute) by the Barring drive to slowly dislodge the

    mill charge.

    The inching drive components include a prime mover, speed reducer, and a

    connection - engaged by hand or automatically, between the inching reducer

    and the main drive. Also included in the system, is a brake or backstop to hold

    the equipment when it is stopped in an unbalanced position as well as other

    appropriate safety devices.

    Starting a mill with a locked load will not necessarily causes visible damage to

    the shell or any other of the mill components, but can cause drive-shaft

  • misalignment and will impact on the long term integrity of the mill. Locked

    charge start-ups will cumulatively reduce the overall mill life.

    As a general rule of thumb all tumbling mills are to be barred for at least 2

    revolutions prior to start-up if they have been standing for more than 6 hours.

    How do they work:The inching drive for the mill motor consists of a small

    motor with reducer (gearbox) and clutch/brake package. During normal

    operation, the clutch/brake is disengaged. However, if forsome reason it

    becomes necessary to turn the motor slowly, or inch the motor, the main

    motor isstopped and the clutch is engaged on the inching package. Using the

    inching drive package, themain motor now can be turned or inched at a much

    slower rate. This task is accomplished by havingthe inching package supply a

    pulsing feature to provide rotational movement through the motorpinion to

    the main motor drive. To return to the main motor, the clutch is disengaged

    and the main motor is energized.

    Typical Barring System

  • 3.BACKGROUND

    At present, most mills are started by the main drive motor from an initial rest

    position. This results in large torque outputs and consumption of high torque

    dependant electric current. Even when Barring Systems are fitted this is the

    case as at the end of the traditional Barring operation the drive train comes to

    rest for the system to be dis-engaged. The main drive motor again is at rest

    and for milling needs to overcome initial inertias from its rest position.

    In a mill configuration with self-disengagement coupling systems this is not the

    case as the main drive motor does not start from initial rest position as the mil

    is rotated by the barring motor before start-up. At start-up the drive motor

    does not start from rest, but from the barring rotation speed.An automatic

    coupling ensures speed dependant uncoupling of the barring system after

    start-up of the main drive motor.

    Siguiri, one of the Anglogoldashantis Continental Africa Region business units

    have a Self-disengagement system installed on a large mill and due to its

    successful history and operational simplicity it was decided to evaluate this as

    possible implementation on other business units.

    4.MILL DETAIL

    Ball Mill : 6.1m diam. 9.0m (Polysius)

    Motor Rating : kW 6000 , 994 r.p.m

    Gearbox Rating : 57kN.m

    Gearbox Type : Flender H3H12, Double Reduction Combiflex

  • 5.PRINCIPLE AND OPERATION OF SELF-DISENGAGEMENT BARRING SYSTEMS

    With the new baring gear coupling installed, the mill motor starts up whilst

    baring is taking place, and the mill load is in motion. The baring gear coupling

    slides back in a keyway and a limit switch simply stops the baring motor. The

    self-disengagement clutch is a directionally actuated freewheel clutch.

    Part A is mounted on the driver gear unit shaft with axial movement by means

    of a key and part B on unit to be driven and is fixed.

    This imply that A is driving B in the Barring mode.

    The overrunning clutch is engaged in the stationary condition by shifting partA

    axially to engage with B.

    Once the speed of clutch part B is higher than that of clutch part A (Starting

    point of main motor), independent dis-engagement caused by the angled faces

    of the engaging dogs on clutch part A and clutch part B takes place. Motor and

    driven machine unit are then dis-engaged mechanically. A is then locked in the

    dis-engaged position.

    The overrunning clutch is suitable only for horizontal arrangement

    B B A

  • COUPLING ENGAGED

    COUPLING DIS-ENGAGED

    A

    B

  • 6.METHODOLOGY

    The methodology was to develop a model and quantify the effect of Self-

    disengagement barring systems on key performance indicators of a mill with

    special attention to life time reliability.

    The purpose of the investigation was to compare the following key

    performance indicators for operation of a ball mill with and without a self-

    disengagement barring system. The following was taken as critical evaluation

    factors.

    Total cost.

    Fatigue life.

    At present, most mills are started by the main drive motor from an initial rest

    position. This results in large torque outputs and consumption of high torque

    dependant electric current.

    The following strategic questions were addressed:

    How can the effect of the barring & automatic smooth coupling be

    modelled on the mill life cycle cost?

    How does a model correspond to reality and why is the correspondence

    justified?

    What is the lifecycle cost saving realised by changing a typical existing

    system into this automatic coupling system?

    What is the effect of the barring system with smooth coupling on the

    fatigue life of the gearbox, motor and the rest of the drive train? How

    will it influence reliability and availability of the system?

    Is it feasible to modify an existing ball mill system into the automatic

    coupled system given remaining service life and other client specified

    parameters?

    This report is complementary to the Financial Excel model and contains the

    workflow of the design process, results and findings, and, user manual thereof.

  • 7.COST MODEL

    As total costs was one of the critical factors and a cost model was developed to

    determine this. The requirement was a model that can be used by the different

    plants to establish the total cost of ownership of its ball mill. The output of the

    model must also enable to quantify the savings achieved by installing a barring

    system on the ball mill.

    The financial model is a representation of all activities related to the ball mill

    that influence its cost of ownership. Such models provide clarity regarding the

    financial consequences of past activities and allow for educated decisions

    regarding future activities. The financial model will also serve as a knowledge

    base to capture important information and statistics.

    The following steps were followed in creating the model:

    The ball mill components and financial variables were identified and

    any relationships quantified.

    Historic and operating data were obtained for all costs associated

    with the identified components and variables. The data included

    sufficient information to determine:

    o Capital costs.

    o Operating costs.

    o Product costs.

    o Repair and maintenance costs.

    Lost production due to unplanned failures

    These user requirements imply that:

    The model must be able to project costs over a fixed period (for example

    20 years) to enable valid comparison of results from different mines.

    There must be a function that allows the user to determine predicted costs

    for the system when a barring system is installed.

    The primary requirements of the model can be summarised as follows:

    The model should run on a platform that is available at all sites to avoid any

    unnecessary expenditure of new software. Excel would be a suitable

    platform.

    Data entry can be manual but must be user friendly.

  • The model must be able to store data that can be easily retrieved for

    calculation purposes, i.e. database capability.

    The graphical user interface (GUI) must be user friendly and easy to

    understand.

    Report writing must be quick and easy and the report may be in Excel or

    Word format.

    The model must be able to calculate Net Present Value for all costs

    accrued to date.

    The model must be able to calculate predicted Cost of Ownership over a

    period of 20 years.

    The model must be able to calculate average operating cost of a ball mill

    per year.

    The model must allow for an option to calculate the effect of installing a

    barring system at a point in time.

    Some calculations will require user input for inflation and interest

    values.

    The model must be applicable to all sites.

    The model must be easy to install and maintain with the use of a user

    manual.

    The model must not interfere with any other software applications that

    are currently run by the client.

    The model must be stable.

    To quantify the financial consequence of a barring system, the following tasks

    were conducted:

    Key measured performance indicators were compared between the

    system during start-up with and without the automatic coupling system.

    The performance indicators were utilized to aid in the construction of a

    fatigue model for the ball mill system.

    A life expectancy model was constructed based on the fatigue model, to

    establish the relationship between the system with and without the

    automatic coupling. This relationship is used to give an indication of the

    expected life of a system without barring, which is then used in the

    construction of the financial model.

  • Examples of Cost model Input-sheets.

    General Mill detail

    Gearbox Detail

  • Maintenance detail

    Operational Detail

  • 8.ADVANTAGES OF SELF-DISENGAGEMENT BARRING SYSTEMS

    Smooth torque transfer through the drive train because backlash is

    taken up by the barring drive.

    Angular momentum at point of start-up of the main drive. The system

    does not start from rest that could result in lower torques.

    Bearings remain lubricated during barring.

    Small and cheap motors and drive keeps the mill in rotation.

    Safe and non-complex system to operate and operator friendly

    Very low and basic maintenance intensive system

    Non-complex retrofit on current systems

    Time saving operation

    Relative small capital layout

    Eliminates the possibility of bridging out limit switches as it form part of

    the Control Philosophy.

    9.DIS-ADVANTAGES OF SELF-DISENGAGEMENT BARRING SYSTEMS

    Capital requirements.

    Retrofit to current systems that could require a specific design.

    In certain cases space can be a problem to retrofit depending on existing

    design.

    Production loss time to install.

    Culture change to long existing practise.

    Resistance to change the not invent here syndrome

  • 10.INSTRUMENTATION and MEASUREMENTS

    The gearbox fitted with the Self disengagement Barring system was

    instrumented as follow:

    Top view of ball mill layout, showing the accelerometer positioning.

    1. Two 90 degree bi-axial shear strain gauges on the barring gearbox

    output shaft. Wired to measure torque output One full-bridge channel

    (position 1)

    2. Two 90 degree bi-axial shear strain gauges on the main gearboxs input

    shaft. Wired to measure torque output One full-bridge channel

    (position 2)

    3. Three accelerometers on the main gearbox housing on the input shaft

    end measuring tri-axial acceleration of the main gearbox.

    4. Three accelerometers on the main gearbox housing on the barring

    coupling end measuring tri-axial acceleration of the main gearbox.

    This totals to 2 strain signals and 6 acceleration signals. The position of the

    accelerometers and strain gauges is shown in the above figure.

    Photo of instrumented barring output shaft and part of main gearbox system.

  • 11.MEASUREMENT RESULTS

  • ACCELERATION

    Acceleration on Gearbox Casing atBarring end. UNCOUPLED(Fig 1)

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acce

    lerat

    ion in

    the

    orth

    ogon

    al dir

    ectio

    ns on

    th

    e ba

    rring

    en

    d w

    ith th

    e ba

    rring

    sy

    stem

    un

    coup

    led

    Acceleration [m/s

    2

    ] in x direction

    Time

    [s]

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acceleration [m/s

    2

    ] in y direction

    Time

    [s]

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acceleration [m/s

    2

    ] in z direction

    Time

    [s]

  • Acceleration on Gearbox Casing atMain drive end. UNCOUPLED (Fig 2)

  • Acceleration onGearbox Casing atBarring end. COUPLED(Fig 3)

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acce

    lerat

    ion in

    the or

    thogo

    nal d

    irect

    ions

    on the

    dri

    ve en

    d with

    the

    ba

    rring s

    yste

    m un

    coup

    led

    Acceleration [m/s

    2

    ] in x direction

    Time

    [s]

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acceleration [m/s

    2

    ] in y direction

    Time

    [s]

    6080

    100

    120

    140

    160

    180

    -3

    -2

    -1012

    Acceleration [m/s

    2

    ] in z direction

    Time

    [s]

  • Acceleration onGearbox Casing at Main drive end.COUPLED (Fig 4)

    020

    4060

    8010

    012

    0-0.

    6

    -0.

    4

    -0.

    20

    0.2

    0.4

    0.6

    0.8

    Acce

    lerat

    ion in

    the or

    thog

    onal

    direc

    tions

    on

    th

    e ba

    rring e

    nd w

    ith th

    e ba

    rring s

    yste

    m co

    upled

    Acceleration [m/s

    2

    ] in x direction

    Time

    [s]

    020

    4060

    8010

    012

    0-2.

    5

    -2

    -1.

    5

    -1

    -0.

    50

    0.51

    1.5

    Acceleration [m/s

    2

    ] in y direction

    Time

    [s]

  • INPUT SHAFT TORQUE

    COUPLED(Fig. 5)

    020

    4060

    8010

    012

    014

    0-1

    -0.500.51

    Acce

    lerati

    on in

    the or

    thogo

    nal d

    irecti

    ons

    on the

    m

    ain dri

    ve en

    d with

    the

    ba

    rring s

    ystem

    co

    upled

    Acceleration [m/s

    2

    ] in x direction

    Time

    [s]

    020

    4060

    8010

    012

    014

    0-1

    -0.500.51

    Acceleration [m/s

    2

    ] in y direction

    Time

    [s]

    020

    4060

    8010

    012

    014

    0-3

    -2

    -1012

    Acceleration [m/s

    2

    ] in z direction

    Time

    [s]

  • COUPLED(Fig 6)

    INPUT SHAFT TORQUE

    UNCOUPLED(Fig 7)

    0 20 40 60 80 100 120 140-15

    -10

    -5

    0

    5

    10

    15

    20

    25

    30

    35

    X: 29.12Y: 32.92

    Time [s]

    Torq

    ue [kN

    .m

    ]

    Torque on main gearbox input shaft with barring system coupled

    4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6-5

    0

    5

    10

    15

    20

    25

    30

    35

    40

    45

    X: 5.051Y: 32.09

    Time [s]

    Torq

    ue [kN

    .m

    ]

    X: 5.152Y: 28.13

    Detail section of initial torque peaks on main gearbox input shaft

  • UNCOUPLED(Fig 8)

    12.RESULTS

    Acceleration results

    60 80 100 120 140 160 180-15

    -10

    -5

    0

    5

    10

    15

    20

    25

    30

    35

    40Torque on main gearbox input shaft with barring system uncoupled

    Time [s]

    Torq

    ue

    [kN.m

    ]X: 85.68Y: 34.65

    61.3 61.4 61.5 61.6 61.7 61.8 61.9 62-5

    0

    5

    10

    15

    20

    25

    30

    35

    40

    45 X: 61.65Y: 38.9

    Detail section of initial torque peak on main gearbox input shaft

    Time [s]

    Torq

    ue

    [kN.m

    ]

    X: 61.73Y: 1.557

    X: 61.7Y: 24.37

    X: 61.71Y: 9.866

  • From Fig. 1 to Fig. 4 the accelerations measured on the main gearbox housing

    is shown. When comparing the two sets of acceleration data the initial

    acceleration magnitude is larger on the gearbox housing (clearly visible on the

    drive end accelerations) when the barring system is uncoupled, which supports

    the statement above.

    Comparing Fig.2 with Fig.4 the acceleration measured on the main drive end

    on the y direction recuses from 0.8 mm/s.s uncoupled to 0.25 mm/s.s coupled.

    Reduction of 68%.

    Torque results

    The rated power of the main motor output of the ball mill is 6MW, with a

    rotational speed of 994rpm.

    PT =

    where:

    P = rated power of the motor, [Watt].

    = angular velocity, [radians].

    T = rated torque, [kN.m].

    Using this equation the rated torque for the gearbox input shaft is thus

    calculated as being 57kN.m. The average operational torque calculated from

    the measured data, over the operational section of the data for the main

    gearbox, was calculated as being 21.2kN.m. This operational torque of the

    main gearbox input shaft is 37% of the rated torque of the system.

    When considering the start-up of the ball mill with the barring system

    uncoupled, (Fig.7&8) impact loads is seen by the shaft and other components

    of the gearbox. Impact loads could create stresses that are significantly higher

    than the stresses created when similar loads are applied gradually. The torque

    data for the main gearbox input shaft shows two significant impact loads, one

    at initial start-up, as shown in the red section and the second at the sorting of

    the soft start system, shown in the green section of Fig. 5 and Fig. 7. These

    impacts will have no influence on the fatigue life of the shafts; however the

    compressive force seen on the gearbox internals is significant.

    The results from the measured data for the ball mill with barring, included:

  • When the mill drive motor is started, it uncouples the barring system

    clutch. To uncouple this clutch the torque on the barring system changes

    due to directional changes that will cause a peak (32.09kN.m).However, the

    peak shown in Fig.8 (38.9kN.m) where the mill is started without barring is

    caused by the backlash in the mill gearbox and is transferred through the

    gearbox. Therefore, for damage comparison on the mill gearbox, the peak

    on the system started with barring (32.09kN.m) may be ignored and the

    peak on the system started without barring (38.9kn.m) shall be included.

    The first torque peak to be considered for the main gearbox input shaft was

    calculated as being 28.13kN.m. This is about a 32.6% increase from the

    average operational torque (21.2kN.m) calculated for the shaft and is 49%

    of the rated torque of the gearbox. (57kN.m).

    The results seen in Fig. 7andFig. 8 for the ball mill with no barring included:

    The first initial torque peak for the main gearbox input shaft was calculated

    as being 38.9kN.m. This is about a83.4% increase from the average

    operational torque (21.2kN.m) calculated for the shaft and is 68% of the

    rated torque of the gearbox (57kN.m).

    Fig 8 showsthe torque peaks with two distinct high torque values. Also, a

    second peak is seen at 24.37kN.m. The initial torque peak was considered

    in the fatigue assessment as this is the highest value calculated and the

    second peak is similar to what was seen by the system with barring and

    would not change the fatigue life when comparing the two systems.

    From the above calculated results the second torque peak in both the barring

    data and the no barring data is almost the same magnitude and the difference

    would have no effect on the operational life of the system, when comparing

    the barring and the no barring data. The result that would have an effect on

    the operational life of the system is the difference in the first torque peak

    (38.9kN.m) in the system uncoupled and the second torque peak (28.13kN.m)

    in the system coupled. This difference equates to a 38% increase in torque

    experienced by the system compared to starting up from the barring speed.

    13.CONCLUSION

  • During the assessment on the data measure Siguiri mine the following was

    determined:

    The torque on the main gearbox input shaft, included:

    o An initial torque peak on both the sets of data, with barring and no

    barring. The initial torque peak in the data with barring could be

    neglected as this torque is not transferred to the gearbox, as

    supported by the acceleration data. The acceleration data showed a

    peak not at the initial start-up.

    o The torque peaks on the input shaft were 38.9kN.m for the system

    with no barring and 28.13kN.m for the system with barring.

    o A second torque peak is seen when the soft start system is

    disengaged. This peaks was not considered in the life assessment of

    the system as this peak was similar in both sets of data and would

    have no effect during comparison of the data.

    The torque on the barring gearbox output shaft reached 7kN.m.

    The fatigue life assessment included the design of a fatigue model for the

    gearbox system along with a model for system availability. The results from

    the effect of barring on fatigue life included:

    The contact stresses in the pinions and gears increase by about 17% for no

    barring compared to barring.

    The fatigue life assessment showed that the shafts are designed for an

    infinite life time and no difference was seen between barring and no

    barring on the fatigue life of this component.

    There is a significant change in operational life cycles for the pinions and

    gears of the gearbox, which are most likely the components to fail. A ration

    of 9.5 x 10-4

    for life cycles of no barring compared to barring was

    determined.

    A reduction in operational life of 31.5% for the gearbox is calculated when

    the barring system is not used on the ball mill system.

    A financial model was constructed according to user requirements to enable

    the comparative analysis of cash flows for a Ball Mill with a barring and

    automatic coupling mechanism and a Ball Mill without it. Utilising this model in

    general indicated a payback of less than a year on a typical 3 Mw Mill.

    14.FINAL WORD

  • It is my absolute opinion that this is what is called a no brainer.

    Just go and do it.

    From a safety perspective and with specific attention to Stored Energy this

    system is almost in comparable to the normal practice. Just the elimination of

    force by hammer or lever to engage the wheel speaks for itself .If you consider

    sometimes the physical condition of the engagement wheel you would prefer

    not to be present at this operation during specifically at night time.

    This is also a system that as far as bridging-out safety systems is concerned

    can be promoted as not possible. This becomes part of the Mill Control

    Philosophy and wired into the starting sequence.

    It is my absolute belief that this is a Step Change in Mill starting procedures

    with many other benefits and advantages not quantified in this paper. This

    does not eliminate other factors to be considered that will hinder this change

    but in no doubt I believe we can overcome that.

    I sincerely hope this paper will inspire either an individual, a team or in

    whatever way to embark on this route.

    15. ACKNOWLEDGEMENT

    Dr. Michiel Heyns from Investmech for the in depth study conducted on the

    field.

    AMRE for the request to write this paper.