METHOD AND APPARATUS FOR PRODUCING A PRIMARY ROLL OF MATERIAL

METHODE ET INSTALLATION POUR PRODUIRE UN ROULEAU PRIMAIRE DE MATERIAU

English Abstract

The method is for producing a first primary roll
having a predetermined lateral surface defined by a diameter
Df. The primary roll is made of material wound around a
spindle. The material is used to produce smaller secondary
rolls of material. The method comprises steps of (a)
calculating a portion Sf of the lateral surface, which is
covered by the spindle; (b) calculating a portion Si of the
lateral surface, which represents material needed to produce
the smaller secondary rolls of material: (c) calculating a
compression factor K1 which is derived from a compression
rate K of a previous second primary roll used to produce
previous secondary rolls with respect to the previous
secondary rolls; (d) calculating Df where:
<IMG>
(e) winding up material around the spindle to produce the
first primary roll until a diameter of the first primary
roll reaches said diameter Df; whereby loss of material is
reduced by taking into account compression factor K1 which
varies with respect to time. An apparatus to perform the
present method is also provided.

Claims

The embodiments of the invention in which an
exclusive property or privilege is claimed are defined as
follows:
1. Method for producing a first primary roll having a
predetermined lateral surface defined by a diameter Df said
primary roll being made of material wound around a spindle,
said material being used to produce smaller secondary rolls
of material, said method comprising steps of:
(a) calculating a portion Sf of said lateral surface,
which is covered by said spindle;
(b) calculating a portion Si of said lateral surface,
which represents material needed to produce said smaller
secondary rolls of material;
(c) calculating a time varying compression factor K1
which is derived from a compression rate K of a previous
second primary roll used to produce previous secondary rolls
with respect to said previous secondary rolls;
(d) calculating Df where:
<IMG>
(e) winding up material around said spindle to
produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor K1 which varies with respect to
time.
2. Method according to claim 1, wherein said compression
rate K = [(sum of lateral surfaces of material of said
previous second primary roll used to produce said previous
secondary roll)/(sum of lateral surfaces of material of said
17

previous secondary rolls)].
3. Method for producing a first primary roll having a
predetermined lateral surface defined by a diameter Df, said
primary roll being made of material wound around a spindle,
said material being used to produce smaller secondary rolls
of material, said method comprising steps of:
(a) calculating a portion Sf of said lateral surface,
which is covered by said spindle;
(b) calculating a portion Sp of said lateral surface,
which represents remaining unusable material wound around
said spindle;
(c) calculating a portion Sa of said lateral surface,
which represents an error margin determined by an operator;
(d) calculating a portion Si of said lateral surface,
which represents material needed to produce said smaller
secondary rolls of material;
(e) calculating a time varying compression factor K1
which is derived from a compression factor K where K = [(sum
of lateral surfaces of material of a previous second primary
roll used to produce previous secondary rolls)/(sum of
lateral surfaces of material of said previous secondary
rolls)];
(f) calculating Df where:
<IMG>
(g) winding up material around said spindle to
produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor K1 which varies with respect to
time.
18

4. Method according to claim 2, wherein step (c) further
comprises steps of:
(i) calculating at least another compression rate K
of at least another previous primary roll with respect to
other previous secondary rolls; and
(ii) calculating an average value of said compression
rates K so that said compression factor K1 is derived from
said average value.
5. Method according to claim 3, wherein step (e) further
comprises steps of:
(i) calculating at least another compression rate K
of at least another previous primary roll with respect to
other previous secondary rolls; and
(ii) calculating an average value of said compression
rates K so that said compression factor K1 is derived from
said average value.
6. Apparatus for producing a first primary roll having a
predetermined lateral surface defined by a lateral diameter
Df, said primary roll being made of material wound around a
spindle, said material being used to produce smaller
secondary rolls of material, said apparatus comprising :
means for calculating a portion Sf of said lateral
surface, which is covered by said spindle;
means for calculating a portion Si of said lateral
surface, which represents material needed to produce said
smaller secondary rolls of material;
means for calculating a time varying compression
factor K1 which is derived from a compression rate K of a
previous second primary roll used to produce previous
secondary rolls with respect to said previous secondary rolls;
19

(d) means for calculating Df where:
<IMG>
(e) means for winding up material around said spindle
to produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor K1 which varies with respect to
time.
7. Apparatus according to claim 6, wherein said
compression rate K = [(sum of lateral surfaces of material
of said previous second primary roll used to produce said
previous secondary roll)/(sum of lateral surfaces of
material of said previous secondary rolls)].
8. Apparatus for producing a first primary roll having a
predetermined lateral surface defined by a lateral diameter
Df, said primary roll being made of material wound around a
spindle, said material being used to produce smaller
secondary rolls of material, said apparatus comprising:
means for calculating a portion Sf of said lateral
surface, which is covered by said spindle;
means for calculating a portion Sp of said lateral
surface, which represents remaining unusable material wound
around said spindle;
means for calculating a portion Sa of said lateral
surface, which represents an error margin determined by an
operator;
means for calculating a portion Si of said lateral
surface, which represents material needed to produce said
smaller secondary rolls of material;

means for calculating a time varying compression
factor K1 which is derived from a compression rate K where
K = [(sum of lateral surfaces of material of a previous
second primary roll used to produce previous secondary
rolls)/(sum of lateral surfaces of material of said previous
secondary rolls)];
means for calculating Df where:
<IMG>
means for winding up material around said spindle to
produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor K1 which varies with respect to
time.
9. Apparatus according to claim 7, further comprising:
means for calculating at least another compression
rate K of at least another previous primary roll with
respect to the previous secondary rolls; and
means for calculating an average value of said
compression rates K so that said compression factor K1 is
derived from said average value.
10. Apparatus according to claim 8, further comprising:
means for calculating at least another compression
rate K of at least another previous primary roll with
respect to other previous secondary rolls; and
means for calculating an average value of said
compression rates K so that said compression factor K1 is
derived from said average value.
21

Description

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209724~
METHOD AND APPARATUS FOR PRODUCING A pRT~ARY ROLL OF
MATERIAL
FIELD OF THE INVENTION:
The present invention is concerned with a method and
an apparatus for producing a primary roll having a
predetermined lateral surface defined by a Diameter Df. The
primary roll is made of material wound around a spindle. The
material is used to produce smaller secondary rolls of
material. More specifically, the present invention can be
used in the paper industry.
RA~Kf~ROUND OF THE INVENTION:
Known in the art, there is the U.S. Patent No.
4,519,039 of Bhupendra S. SURANA et al, granted on May 21,
1985 in which there is described a programmable controller
including coil diameter calculator, strip speed derivation
and inertia compensation. The controller is associated with
a reel system for the generation in normalized digital form
of a coil diameter of the reel instantaneously to allow
initial calibration between successive coil winding and
unwinding operations and automatic generation of a current
reference for reel motor drive control.
Also known in the art, there is the U.S. Patent No.
4,631,682 of David T. NG et al, granted on December 23,
1986, in which there is described a control system which
provides automatic control of winder deceleration and
stopping to a preset sheet length, or preset roll diameter.
The system utilizes a closed loop control of drive
deceleration and automatic compensation for layers slabbed
off following a sheetbreak.
Also known in the art, there is the U.S. Patent No.

~0972i~
5,086,984 of Douglas E. TUREK et al, granted on February 11,
1992, in which there is described a method of predicting
final yarn package diameter during winding of yarn onto the
package. The yarn is to be wound onto the package for a
known period of time to obtain the final yarn package
diameter. The method comprises the steps of: measuring the
time for the package to grow to a known diameter, and
predicting yarn package diameter using a predetermined
correlation.
Also known in the art, there are the U.S. Patent No.
4,913,366; 4,883,233; 4,811,915; 3,910,516; and 3,792,820
which describe different apparatuses and methods relating to
the production of a rolI of material.
In the paper industry, big primary roll are used to
produce smaller secondary rolls which will be sold to
clients. When successive primary rolls are used to produce
secondary rolls, the amount of paper wound around each
primary roll with identical diameter will not produce the
same amount of material on secondary rolls because the
compression rate of the paper wound around each primary roll
with respect to material wound around their respective
secondary rolls varies from time to time because the
operating conditions of the machines used to produce
secondary rolls from a primary roll are not exactly the same
from time to time.
Accordingly, to solve this problem, it is known to
wound around each primary roll an additional amount of paper
to be sure that there will be enough paper for the secondary
rolls that should be produced.
One problem with this is that a certain amount of
paper is lost at the end of each primary roll when it is
unrolled.

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2097246
None of the above patents provides a method or an
apparatus that takes into account the fact that the
compression rate at which the paper is wound around a
primary roll with respect to secondary rolls is not
constant.
It is an object of the present invention to provide
a method and an apparatus for estimating with more precision
the final diameter of the primary roll so that the loss of
material when said primary roll is used to produce secondary
rolls is reduced to minimum.
SUMMARY OF THE INVENTION:
According to the present invention, there is provided
a method for producing a first primary roll having a
predetermined lateral surface defined by a diameter Df said
primary roll being made of material wound around a spindle,
said material being used to produce smaller secondary rolls
of material, said method comprising steps of:
(a) calculating a portion Sf of said lateral surface,
which is covered by said spindle;
(b) calculating a portion Si of said lateral surface,
which represents material needed to produce said smaller
secondary rolls of material;
(c) calculating a compression factor K1 which is
derived from a compression rate K of a previous second
primary roll used to produce previous secondary rolls with
respect to said previous secondary rolls;
(d) calculating Df where:
Df = ~/(4 (Sf ~ (SiKl) ) ) /1~
(e) winding up material around said spindle to
.~

20 97246
produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor Kl which varies with respect to
time.
Also according to the present invention, there is
provided an apparatus for producing a first primary roll
having a predetermined lateral surface defined by a lateral
diameter Df, said primary roll being made of material wound
around a spindle, said material being used to produce
smaller secondary rolls of material, said apparatus
comprising :
means for calculating a portion Sf of said lateral
surface, which is covered by said spindle;
means for calculating a portion Si of said lateral
surface, which represents material needed to produce said
smaller secondary rolls of material;
means for calculating a compression factor K1 which
is derived from a compression rate K of a previous second
primary roll used to produce previous secondary rolls with
respect to said previous secondary rolls;
(d) means for calculating Df where:
Df = ~/(4 (Sf + (SiKl) ) ) /rc
(e) means for winding up material around said spindle
to produce said first primary roll until a diameter of said
first primary roll reaches said diameter Df;
whereby loss of material is reduced by taking into account
said compression factor Kl which varies with respect to
time.
The objects, advantages and other features of the
present invention will become more apparent upon reading of
A

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20972~6
the following non restrictive description of a preferred
embodiment thereof given for purpose of exemplification only
with reference to the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS:
Figure 1 is a schematic diagram illustrating
schematically how a primary roll is produced, and how a
secondary roll is produced from a primary roll;
Figure 2 is a schematic diagram illustrating with
more details a working station shown in figure 1; and
Figure 3 is a flow chart diagram illustrating the
method for producing a primary roll in accordance with the
present invention.
DETATT.~n DESCRIPTION OF THE DRAWINGS:
Referring now to Figure 1, there is shown working
station 2 where a sheet of paper 4 is wound around a metal
spindle 6 to produce primary roll 8. Also, there is shown
working station 10 where primary roll 9 is unrolled to
produce secondary roll 12. From one primary roll 9, several
smaller secondary rolls 12 are produced to be delivered to
clients. The final diameter Df of the primary roll 9 depends
directly on the final diameter of the secondary rolls 12 to
be delivered to the clients.
It has been found that for an identical combination
of secondary rolls to be produced from a primary roll, the
diameter of successive primary rolls will not be the same.
This is caused by the variation of volumetric reduction of
the paper sheet 14 due to mechanical work at the working
station 10 when the paper sheet 14 is unrolled from the
primary roll 9 and wound around spindle 13 to produce
secondary roll 12.

2097246
Also , when primary roll 8 is produced, paper sheet
4 can be torn or a portion of paper sheet 4 can have an
unacceptable quality. All these factors have to be taken
into consideration so that the primary roll 8 has a
sufficient amount of paper to produce predetermined
secondary rolls to be delivered to clients.
The working stations 2 and 10 are provided with
several equipments which comprise a computer 16, a terminal
18 disposed nearby an operator, an optical detector 20 for
detecting the number of turns made by drum 22, an optical
detector 24 for detecting the number of turns made by
secondary roll 12, and another optical detector 26 for
detecting the number of turns made by spindle 28.
Referring now to Figure 2, there is shown with more
details working station 2. Paper sheet 4 coming from a paper
machine (not shown) is moved around drum 30 to be wound
around spindle 6. Spindle 6 is supported by means of rails
32. A constant pressure is applied on each side of the
spindle 6 of primary roll 8 by means of cylinders 34. Only
one cylinder 34 is shown in this figure, but it is
understood that each side of spindle 6 is subjected to a
pressure applied by a cylinder. Paper sheet 4 is wound
around spindle 6 until the diameter of primary roll 8
reaches a predetermined value.
The present system is capable of measuring the
diameter of primary roll 8 in real time. Several known
methods can be used for measuring this diameter. According
to working station 2, shown in Figure 2, the diameter of
primary roll 8 is calculated from pulses received from
detector 40 and detectors 42. Only one detector 42 is shown
in Figure 2, but the other side of spindle 6 is also
provided with a detector. Detector 40 generates a pulse

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20972~6
during each turn of drum 30 and detectors 42 generate a
pulse during each turn of spindle 6. A reflecting sticker 44
is stuck at each end of spindle 6 and is used to reflect an
optical ray generated by detectors 42. When one detector 42
receives a reflection from its corresponding sticker 44, it
generates instantaneously an electric pulse which is sent to
computer 16. Only one of detectors 42 is used at the time.
The second detector 42 is used as a back-up. Computer 16
measures with precision the period of time between pulses
generated by detectors 40 and 42 and calculates in real time
the radius D of primary roll 8.
D = [(Tf De)/Te], where D is the diameter of primary
roll 8, Tf is the period of time measured between two pulses
generated by detector 42, De is the diameter of drum 30, and
Te is the period of time measured between two pulses
generated by detector 40.
Also shown in this Figure 2, there are a display 50
showing the period of time remaining before the actual
diameter of primary roll 8 reaches a predetermined diameter,
an alarm 52, a detector 54 detecting when paper sheet 4 is
torn up, a button 56 by which the operator can also indicate
to computer 16 that paper sheet 4 is torn up, another button
58 by which the operator can indicate to computer 16 that
quality of paper is not acceptable, and a pressure detector
60 by which computer is informed of pressure applied by
cylinders 34.
Referring now to Figures 1 and 2, it is understood
that diameters of secondary roll 12 and primary roll 9 of
working station 10 can be determined in real time by optical
means similar to the ones shown in Figure 2.
The apparatus for producing first primary roll 8
having a predetermined lateral surface defined by lateral

20972~6
diameter Df is shown in Figures 1 and 2. The primary roll 8
is made of material wound around spindle 6. The material is
used to produce smaller secondary rolls 12 of material. The
apparatus comprises means for calculating a portion Sf of
the lateral surface of primary roll, which is covered by its
spindle. This means for calculating is performed by computer
16 and the calculation is done with respect to parameters
entered by the operator by means of terminal 18.
The apparatus also comprises means for calculating a
portion Sp of the lateral surface, which represents remain-
ing unusable material wound around the spindle of primary
roll. Again, this means for calculating is performed by com-
puter 16 with respect to parameters entered by the operator.
The apparatus also comprises means for calculating a
portion Sa of the lateral surface, which represents an error
margin determined by the operator. Again, this error margin
corresponds to parameters entered in computer 16 by the
operator.
The apparatus also comprises means for calculating a
portion Si of the lateral surface which represents material
needed to produce several smaller secondary rolls of
material. This portion Si is calculated from parameters
entered by the operator in computer 16.
Also, the apparatus comprises means for calculating
a compression factor Kl which is derived from a compression
rate K where K = [(sum of lateral surfaces of material of
previous primary roll 9 used to produce previous secondary
rolls 12)/(sum of lateral surfaces of material of said
previous secondary rolls 12)]. This means for calculating is
performed by computer 16 by means of equipments at working
station 10.
The apparatus also comprises means for calculating Df

2~9724~
where:
Df = ,~ ( 4 ( Sf +Sp+Sa+ ( Si Kl ) ) ) / lC
This means for calculating Df is performed by
computer 16.
The apparatus also comprises means for winding up
material around spindle 6 to produce primary roll 8 until
its diameter reaches diameter Df. This means for winding up
is situated at working station 2. By means of the present
apparatus, the loss of material is reduced by taking into
account compression factor K1 which varies with respect to
time.
Also, the apparatus preferably comprises means for
calculating at least another compression rate K of at least
another primary roll with respect to previous secondary
rolls, and means for calculating an average value of the
compression rates K so that the compression factor K1 be
derived from the average value. Again, the above-mentioned
means for calculating are performed by the computer 16 when
successive primary rolls 9 are unrolled to produce secondary
rolls 12 at working station 10.
In operation, first, when no compression rate K has
been calculated, the operator determines, in an empirical
manner, the diameter of primary roll 8 in function of the
number and the size of secondary rolls to be delivered to
clients. He also adds a security margin. Once primary roll
8 has a diameter which reaches the predetermined diameter,
the operator transfers primary roll 8 from working station
2 to working station 10 where said primary roll becomes
primary roll 9.
Then, paper sheet 14 is engaged around metal spindle
~.
A

2097246
13 so that primary roll 9 be unrolled to produce a first
secondary roll 12. When first secondary roll 12 reaches a
desired diameter, it is removed from working station 10, and
paper sheet 14 is disposed around another spindle 13 to
produce another secondary roll 12. This operation is
repeated until primary roll 9 has not enough paper to
produce another secondary roll 12. Then, the remaining
amount of paper around spindle 28 is lost.
But, as primary roll 9 is unrolled to produce
secondary rolls, a compression factor Kl is derived from a
compression rate K where K = [(sum of lateral surfaces of
material of primary roll 9 used to produce secondary rolls
12)/(sum of lateral surfaces of material of secondary rolls
12)]. The value of the compression factor Kl can be equal to
the compression rate K or it can correspond to an average
value of compression rates K calculated during successive
unwinding of primary rolls 9.
When a value of compression factor Kl has been
obtained, then it is possible to perform the method
according to the present invention for producing the next
primary roll 8 of material according to the flow chart shown
in Figure 3. The method is for producing a primary roll 8
having a predetermined lateral surface defined by a diameter
D~. The material will be used to produce smaller secondary
rolls 12 of material. The method comprises steps of
calculating a portion S~ of the lateral surface, which is
covered by spindle 6; calculating a portion Sp of the lateral
surface, which represent remaining unusable material wound
around spindle 6, such portion Sp being determined by the
operator; calculating a portion Sa of the lateral surface,
which represents an error margin determined by the operator;
calculating a portion S1 of the lateral surface, which

2097246
represents material needed to produce smaller secondary
rolls 12 of material; calculating the compression factor K1
which is derived from the compression factor K defined
earlier; calculating
Df = ~/(4(Sf+Sp+Sa~(SI Kl)))/~
and winding up material around spindle 6 to produce first
primary roll 8 until its diameter reaches diameter Df,
whereby loss of material is reduced by taking into account
pressure factor K1 which varies with respect to time.
The compression factor K1 is calculated in real time
each time that primary roll 9 is unrolled at working station
10. The measure of diameter of primary roll 9 and secondary
roll 12 can be done by means of different optical means,
mechanical means and electrical means. We will now describe
one manner to determine the diameters of rolls 9 and 12. By
means of pulse generator 20 having a resolution of several
pulses by turn, attached to drum 22, and by means of
another pulse generator 26 having a resolution of one pulse
by turn, attached to spindle 28, it is possible to calculate
in real time the diameter of primary roll 9 at working
station 10.
computer 16 calculates diameter Dp of primary roll g
by means of the following equation:
Dp = [(PPT1 Dt)/RT1]
where RT1 is the resolution of pulse generator 20 in pulses
by turn, Dt is the diameter of drum 22, PPT1 is the number
of pulses produced by pulse generator 20 for each pulse
generated by pulse generator 26. Calculation of the diameter
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2097246
of secondary roll 12 is done in a similar manner by using
pulse generators 20 and 24. When secondary roll 12 has been
completed, computer 16 calculates lateral surface of rolls
9 and 12 by means of the following equations:
Sp = [((~(Dp at the beginning)2)/4)~ (Dp at the stop)2)/4)]
Ss = [((~(Ds at the stop)2)/4)-((~(Ds at the beginning)2)/4)]
where Sp is the lateral surface of material of primary roll
9, used for producing secondary roll 12; Dp are diameters at
the beginning and at the stop of primary roll 9 when winding
of secondary roll 12 begins and ends; Ss is the lateral
surface of material of secondary roll 12; and Ds are
diameters at the beginning and at the stop of secondary roll
12.
If three secondary rolls are produced from one
primary roll 9, then:
[ (Sp1 + Sp2 + Sp3)/(SS1 + SS2 + Ss3) ] -
It has to be noted that only the lateral surfaces
transferred from primary roll 9 to secondary rolls 12 are
used in the above-mentioned calculation. Thus, Sp1 is the
lateral surface removed from primary roll 9 during the
winding of secondary roll 12, which has been used for
producing Ss1 of secondary roll 12.
The number and the size of secondary rolls to be
produced from primary roll 9 are entered by the operator in
computer 16 by means of terminal 18. Then, it is possible to
calculate in real time Df of the next primary roll 8 at
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2097246
working station 2 by taking into consideration the
compression factor Kl calculated by computer 16. The
calculation of Df can be done according to the equation
mentioned earlier.
In order to better understand the method according to
the present invention, we will now describe an example with
possible parameters. First, we have to calculate a first
value of Kl when primary roll 9 is unrolled to produce
smaller secondary rolls 12. In this example, four secondary
rolls are produced. Each of the secondary rolls has a
spindle having a diameter of O.lOOm, and has a final
diameter of l.OOm.
For the production of the first secondary roll, DSl
and Dpl are O.lOOm and 2.117m at the beginning, and l.OOm
and 1.864m at the stop. Then, computer 16 calculates S
which is:
[((n(l.OOOm)2)/4)-((n(O.lOOOm)2)/4)] = 0~7775m2.
We also calculate Spl which is:
[((n(2.117m)2)/4)-((n(1.862m)2)/4)] = 0.7969m2.
For the production of the second secondary roll, DS2
and Dp2 are l.lOOm and 1.862m at the beginnin~ and l.OOm and
1.566m at the stop. Then, we calculate Ss2 which is:
[((7r(1.000m)2)/4)-((7r(0.100m)2)/4)] = 0.7775m2.
We can also calculate Sp2 which is:
[((n(1.862m)2)/4)-((n(1.566m)2)/4)] = 0.7969m2.

209 724~
For the production of the third secondary roll, Ds3
and Dp3 are O.lOOm and 1.566m at the beginning, and l.OOm
and l.l99m at the stop. We can now calculate Ss3 and Sp3
with the equations mentioned above and we found that Ss3 =
50.7775m2 and Sp3 = 0.7969m2.
For the production of the fourth and last secondary
roll, Ds4 and Dp4 are O.lOOm and l.l99m at the beginning,
and are l.OOOm and 0.650m at the stop. By using the
equations mentioned above, we found that Ss4 =1.555m2 and
10Sp4 = 0.7969m .
We now calculate K which is in the present case K1.
K = [(0.7969m2 + 0.7969m2 + 0.7969m2 + 0.7969m2)/(0.7775m2
+ 0.7775m2 + 0.7775m2 + 0.7775m2)] = 1.025.
We are now ready to evaluate the final diameter Df of
15the next primary roll 8. First, the operator entered by
means of the terminal 18 a new order for producing a primary
roll 8 at the working station 2, which will have enough
paper to produce three smaller secondary rolls, each of the
secondary rolls having a spindle diameter of O.lOOm and a
20final diameter of 1.2m.
The spindle 6 mounted at the working station 2 has
around it useless paper having a thickness of 0.025m.
Accordingly, a quantity of paper equivalent to this useless
paper has to be added to obtain enough paper for producing
25the three secondary rolls.
The following parameters are entered by the operator
at the terminal: diameter of the spindle 6 of primary roll
8, which is 0.600m; thickness of the useless paper present
around the spindle 6, which is 0.025m; final diameters of
30three secondary rolls to be produced from this primary roll,
each final diameter of the secondary rolls being 1.200m,
diameter spindle of secondary rolls, which is O.lOOm; and a
security margin determined by the operator, which is 0.020m.
14
A

2097246
Also, it has to be noted that the value of K1 is in the
memory of the computer and has a value of 1.025.
First, we calculate Sf which is:
[(~(diameter of the spindle)2)/4],
[ (~r(0.600)2)/4] = 0.283m2.
Then, we calculate Sp which represents the useless
paper having a thickness of 0.025m from the surface of the
spindle. It is known that the spindle diameter is 0.600m,
and the external diameter of the paper loss is:
[(0.025m x 2) + 0.600m]=0.650m.
Sp can now be calculated, which is:
[((~(external diameter of useless paper)2)/4)
((~(internal diameter of useless paper)2)/4)],
[((~(0.650m)2)/4) - ((n(0.600m)2)/4)] = 0.049m2.
We now calculate the surface of paper relating to the
error margin. As the error margin is O.OlOm, we can
evaluate that the internal diameter of the error margin is
0.0650m and its external diameter is 0.670m. The surface
relating to the error margin S is:
[((~(external diameter of the paper relating to the
error margin) )/4) - ((~(internal diameter of the paper
relating to the error margin)2)/4)],
[ ( (7r(0.670m)2)/4) - ( (7r(0.650m)2)/4) ] = 0.021m2 .

2097246
We have now to calculate the surface Si of paper
relating to the production of three secondary rolls, each of
the secondary rolls having a final diameter of 1.200m and a
spindle diameter of O.lOOm. The surface Sbs of one
secondary roll is:
[((~(external diameter of the roll)2)/4)
((~(internal diameter of the roll)2)/4)],
[((7r(1.200m)2)/4) - ((7r(0.100m)2)/4)] = 1.123m2.
Consequently, the surface S. of three secondary rolls
is (2.246m2 x 3) = 6.739m2.
We now calculate the final diameter of the next
primary roll, which is:
Df = ~/(4(Sf + Sa + Sp + (Sl Kl)))/
Df = ~/(4(0.283m2 + 0.021m2 + 0.049m2 + (3.369m2 X 1.025) ) )/7
Df = 2.201m
Computer 16 will now monitor in real time the winding
of the next primary roll 8 at working station 2 and will
stop the winding when the diameter of primary roll 8 will
reach the value of Df.
Although the invention has been described above in
detail in the framework of a preferred embodiment, it should
be understood that the scope of the present invention is to
be determined by the appended claims.

Representative Drawing