EP1240955B1 - Method and apparatus for calculating the roll gap contour - Google Patents
Method and apparatus for calculating the roll gap contour Download PDFInfo
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- EP1240955B1 EP1240955B1 EP20020001665 EP02001665A EP1240955B1 EP 1240955 B1 EP1240955 B1 EP 1240955B1 EP 20020001665 EP20020001665 EP 20020001665 EP 02001665 A EP02001665 A EP 02001665A EP 1240955 B1 EP1240955 B1 EP 1240955B1
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B37/00—Control devices or methods specially adapted for metal-rolling mills or the work produced thereby
- B21B37/28—Control of flatness or profile during rolling of strip, sheets or plates
- B21B37/42—Control of flatness or profile during rolling of strip, sheets or plates using a combination of roll bending and axial shifting of the rolls
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B13/00—Metal-rolling stands, i.e. an assembly composed of a stand frame, rolls, and accessories
- B21B13/02—Metal-rolling stands, i.e. an assembly composed of a stand frame, rolls, and accessories with axes of rolls arranged horizontally
- B21B2013/025—Quarto, four-high stands
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B2261/00—Product parameters
- B21B2261/02—Transverse dimensions
- B21B2261/04—Thickness, gauge
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B2265/00—Forming parameters
- B21B2265/12—Rolling load or rolling pressure; roll force
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B2267/00—Roll parameters
- B21B2267/24—Roll wear
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B37/00—Control devices or methods specially adapted for metal-rolling mills or the work produced thereby
- B21B37/28—Control of flatness or profile during rolling of strip, sheets or plates
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
- B21B38/00—Methods or devices for measuring, detecting or monitoring specially adapted for metal-rolling mills, e.g. position detection, inspection of the product
- B21B38/10—Methods or devices for measuring, detecting or monitoring specially adapted for metal-rolling mills, e.g. position detection, inspection of the product for measuring roll-gap, e.g. pass indicators
- B21B38/105—Calibrating or presetting roll-gap
Definitions
- the present invention relates to a method for calculating the roll gap contour in a roll stand consisting of two work rolls and an associated device.
- a method for online calculation of the roll gap contour is e.g. from US 5,873,277 A out.
- this method is also based on an approximation in that the rollers of the roll stand can be broken down into several disks for the calculation and in between is interpolated.
- This method therefore has a limited accuracy because the Slices cannot be made arbitrarily thin, otherwise the computing time increases increases rapidly and the method is therefore no longer suitable for real-time application would.
- the aim of the present invention is therefore to provide a method and an apparatus for Calculation of the roll gap contour to indicate that gives very accurate results and can still run fast enough for real-time applications.
- the object is achieved in that the roll gap contour online from the Results of a preliminary calculation of the roll gap contour and a online calculation of the roll gap contour is put together.
- This procedure enables the advantages of a pre-calculation, the high accuracy, and an online calculation, the high speed, in one process unite, which makes it possible to form the roll gap contour very quickly and with high accuracy to calculate.
- This procedure is therefore simple in a real time application, e.g. control of a rolling mill.
- the high accuracy of the process the quality of the rolled product can be increased, since predetermined Rolled profiles can be adhered to very precisely.
- a very favorable method results if the solution of the preliminary calculation is calculated using a finite element calculation, since these calculations are very precise and thus an exact solution to the problem is determined. To achieve a certain accuracy, it is sufficient to calculate the first N T Fourier modes for the solution.
- the solution of the preliminary calculation can be calculated as the sum of a solution of a finite element calculation and a solution of a semi-analytical calculation. The solution of the semi-analytical calculation can easily be found if the solution is calculated for an infinitely long cylinder. In order to achieve a certain accuracy, it is advantageous here to calculate the first N F Fourier modes of the solution with the finite element calculation and the N F +1 to N T Fourier modes of the solution with the semi-analytical calculation.
- a further transformation results in a suitable transformation the two-dimensional contact problem to a one-dimensional contact problem, whereby with the solution from the preliminary calculation, the roll gap contour online using the one-dimensional contact problem between contacting rollers and / or between the work roll and the rolling material is calculated and the one-dimensional solution in Connection to the two-dimensional solution is transformed back.
- the nonlinear contact problem can be solved very advantageously iteratively by linearization become.
- the Calibrate a roll stand the roll deformations of a number w rolls of Roll stand calculated directly from the resulting w-1 coupled contact problems become.
- a correction to the strip exit thickness can be made at least one roll stand from the difference in roll deformation during calibration and can be calculated in real time in the conventional rolling process and the Strip exit thickness can be corrected in real time if necessary by changing manipulated variables.
- the comparison of the calculations when calibrating and when conventional rolling process the measured scaffold suspension characteristic at the working point Getting corrected.
- a simple one Check the tolerance tolerance. Because the results of the procedure are very accurate, the quality of the rolled products, through improved tolerance or by adhering to tighter tolerances, which can be further improved Consequence of course also has a positive economic impact.
- the method according to the invention is very advantageous in a higher-level control integrated a rolling mill, the roll gap contour, and possibly the Strip exit thickness, calculated in real time, compared with a specified value and deviations of the roll gap contour or the lying outside the specified tolerance Corrected strip thickness in real time by changing manipulated variables. So you have that Possibility to exactly close the rolled profile from the first to the last roll stand Taxes.
- the settings of the individual roll stands can be coordinated and the quality of the rolled product can be further improved.
- r , ⁇ , z denote the cylindrical coordinates in the radial, angular and axial directions, u the deformation vector field u ( r , ⁇ , z ) with components in the direction r , ⁇ , z , E the modulus of elasticity, ⁇ the transverse expansion factor, ⁇ the density of the Roll material, g the acceleration of gravity, ⁇ ij the tension tensor and u ij the deformation tensor.
- ⁇ rr , ⁇ rz , ⁇ r ⁇ are to be replaced by ⁇ nn , ⁇ nt , ⁇ n ⁇ , where ⁇ nn , ⁇ nt , ⁇ n ⁇ are the components of the stress tensor in transformed coordinates, n denotes the normal direction with respect to of the bearing cone, t is the corresponding transverse direction.
- Each solution L of the Lamé equation can be represented in a generally known form as a Fourier series.
- L FEM + L ANL is chosen for the solution.
- L FEM / n ( r , z ) consequently denotes the nth Fourier mode of a solution of the Lamé equation, calculated using the finite element method in the Fourier space
- L ANL / n ( r , z ) denotes the nth Fourier mode a solution of the Lamé equation, calculated using semi-analytical methods for an infinitely long cylinder.
- the entire solution L is thus constructed from a finite element solution L FEM and a semi-analytical solution L ANL .
- the solutions L FEM of the finite element calculation can be found using the well known methods of finite elements.
- the exemplary solution is therefore only outlined in broad outline.
- the Lamé equation is first multiplied by a test function v and then integrated over the volume V with surface O.
- K F ⁇ ⁇
- F F ⁇ ⁇
- F F ⁇ ⁇
- u the sum of suitable test functions u i / k cos ( n ⁇ ) or u i / k sin ( n ⁇ ) with amplitudes c i / kn and integration via the angle variable ⁇ leads to a linear in each Fourier mode System of equations for determining the amplitudes c i / kn .
- p n (1, z ) 1 for - c 0 ⁇ z ⁇ c 0
- p n (1, z ) 0 otherwise selected.
- the value for c 0 can be chosen arbitrarily.
- the standardized solution L ANL / c 0 ⁇ 0 is thus calculated once and the transformation of the solutions to the actual geometry is carried out only when required. It is of course also possible to determine these solutions L ANL / c ⁇ directly for the respective geometry of the roll. However, the calculation of these solutions L ANL / c ⁇ takes up a lot of computing time, which is why it is cheaper to carry out the calculations only once and then only to carry out the transformations if necessary.
- the work roll A is in contact with the rolling material M on the underside of the bale over the strip width B and in contact with the backup roll S in the top.
- bending forces F B act on the work roll.
- the position of the work roll A can have a displacement d A transverse to the rolling direction.
- the position of the rolling material M can also be shifted transversely to the rolling direction, shift d M.
- the rolling force F W acts between the rolling material M and the work roll A.
- the support roller S is in contact with the work roller A on its underside.
- the stator forces F S act in the bearings of the support roller S.
- Both work roller A and back-up roller S are ground (s A , s S ), both rollers are thermally expanded (t A , t S ) and their contour is changed by wear (v A , v S ).
- These influences can be regarded as known and can either be determined directly from measurements, or in turn come from suitable model calculations.
- the contact area K AM between work roll A and rolling material M, based on the coordinate system of work roll A, is described as follows:
- the contact angle ⁇ k results from the maximum contact length L k with respect to z, divided by the radius of the work roll A.
- the contact area K AS between work roll A and backup roll S, based on the coordinate system of work roll A, is described analogously as follows:
- ⁇ S is again the maximum contact angle with regard to z and L BA or L BS are the bale lengths of work roller A and support roller S.
- the indices i and j therefore describe the point of application of the pressure p and the indices k and I describe the location where the deformation occurs.
- Q AS / ij 1 for ( ⁇ , z ) ⁇ R AS ij .
- x 0 and x 1 describe an additional vertical displacement or tilting of the work roll A.
- the total deformation on the underside of the work roll u kl corresponds exactly to the roll gap contour sought, ie the determination of the roll gap contour is equivalent to the calculation of u kl .
- r k is the non-deformed "distance between work roller A and support roller S.
- the total force and total moment must disappear, which provides two further equations for determining x 0 and x 1 .
- Q AS / ij , x 0 and x 1 are now determined from the contact problem formulated above. It should also be noted that if the pressure distribution p AM / ij between work roll A and rolling material M can be assumed to be known, q AS / ij , x 0 and x 1 can be calculated directly. If the pressure distribution p AM / ij is not known, p AM / ij , q AS / ij , x 0 and x 1 must be calculated iteratively. With the q AS / ij , x 0 and x 1 or p AM / ij thus determined, the roll gap contour, ie u kl , can now be calculated.
- the contact problem formulated above in the form of two inequalities is non-linear, which is why the solution is done iteratively.
- the starting point for the iterative solution can be, for example, a linearized system of equations of the form his.
- the matrix M is formed by the components o AS ij, kl and s AS ij, kl of the relationships for o kl and s kl .
- the vector b then contains all the other components of the relationships for o kl and s kl .
- a significant reduction in computing time can be achieved by reducing the two-dimensional contact problem to a one-dimensional one using a suitable mathematical transformation.
- the offline calculation is first carried out once.
- the standardized semi-analytical solution L ANL / c 0 ⁇ 0 is calculated once (computing time about 20min; Note: all information on the computing time are exemplary and refer to a PC with a clock frequency of 350MHz).
- the Fourier finite element solutions and the radial deformation fields are determined in advance as a preliminary calculation (computing time approximately 40 seconds per roll).
- the online calculations are then carried out in the company as required.
- the deformation of the rolls which occurs is calculated (computing time approximately 1 second per roll stand).
- the roll gap contour can be calculated for each individual roll stand 1 at any time and compared with predetermined values. If deviations are determined by influencing certain manipulated variables, such as the bending force, the roll adjustment or the work roll displacement, the control can make the necessary corrections.
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- Engineering & Computer Science (AREA)
- Mechanical Engineering (AREA)
- Control Of Metal Rolling (AREA)
Description
Die vorliegende Erfindung betrifft ein Verfahren zur Berechnung der Walzspaltkontur in einem Walzgerüst, bestehend aus zwei Arbeitswalzen, sowie eine zugehörige Vorrichtung.The present invention relates to a method for calculating the roll gap contour in a roll stand consisting of two work rolls and an associated device.
Die genaue Kenntnis der Walzspaltkontur in Walzstraßen ist Voraussetzung zur präzisen
Steuerung bzw. Regelung von Bandprofil und Bandplanheit, zwei wesentliche Parameter der
Qualität eines Walzproduktes. In Warmwalzwerken, bestehend aus Vorstraße, Fertigstraße
bzw. Grobblechstraße, können sowohl Band- bzw. Blechprofil als auch Band- bzw.
Blechplanheit kontrolliert werden, in Kaltwalzwerken ist die Bandplanheit die zu
kontrollierende Größe.
In einer mehrgerüstigen Warmwalzstraße ist das korrekte Bandprofil in den ersten
Walzgerüsten einzustellen, während die Bandplanheit des Walzprodukts im letzten Gerüst zu
erzielen ist. Zunächst muss dazu das relative Walzspaltprofil in den ersten Walzgerüsten auf
das relative Zielprofil des Bandes gesetzt werden, weiters ist das relative Walzspaltprofil im
letzten Gerüst auf das relative Walzspaltprofil in den ersten Walzgerüsten abzustimmen. Um
Bandplanheit des Bandes zwischen den Walzgerüsten zu gewährleisten (ruhiger Bandlauf),
ist das relative Walzspaltprofil von Walzgerüst zu Walzgerüst konstant zu halten. Dies
verdeutlicht, wie wichtig die Kenntnis der Walzspaltkontur für die Qualität des Walzproduktes
ist.
Die bisher verfügbaren Methoden zur Ermittlung der Walzspaltkontur lassen sich grob in
zwei Klassen unterteilen:Precise knowledge of the roll gap contour in rolling mills is a prerequisite for precise control or regulation of the strip profile and strip flatness, two essential parameters of the quality of a rolled product. In hot rolling mills, consisting of roughing train, finishing train or heavy plate mill, both strip or sheet profile and strip or sheet flatness can be checked, in cold rolling mills, strip flatness is the size to be checked.
In a multi-stand hot rolling mill, the correct strip profile must be set in the first stands, while the strip flatness of the rolled product must be achieved in the last stand. First of all, the relative roll gap profile in the first roll stands must be set to the relative target profile of the strip. Furthermore, the relative roll gap profile in the last stand must be matched to the relative roll gap profile in the first roll stands. In order to ensure strip flatness between the roll stands (smooth strip run), the relative roll gap profile from roll stand to roll stand must be kept constant. This illustrates how important knowledge of the roll gap contour is for the quality of the rolled product.
The previously available methods for determining the roll gap contour can be roughly divided into two classes:
Da diese offline, d.h. nicht während des eigentlichen Walzvorganges, durchgeführt werden, sind diese Methoden zeitunkritisch. Es werden hier typischerweise Finite-Element-Methoden eingesetzt, bei denen ein Walzgerüst und Walzmaterial mit Finite-Elemente modelliert wird und die Deformation der Walzen unter einer vorgegeben Belastung ermittelt wird. Ein solcher Berechnungsvorgang liefert sehr genaue Ergebnisse, benötigt jedoch einige Minuten bis einige Stunden, wodurch diese Methoden absolut ungeeignet für Echtzeit-Anwendungen, wie z.B. eine Regelung einer Walzstraße, sind. Darüber hinaus können damit natürlich keine dynamischen Einflüsse berücksichtigt werden, da die Lösung nur für die Berechnung mit den vorgegebenen Randbedingungen Gültigkeit hat.Since this is offline, i.e. not carried out during the actual rolling process these methods are not time-critical. There are typically finite element methods here used where a roll stand and rolling material with finite elements is modeled and the deformation of the rollers under a given Load is determined. Such a calculation process provides very accurate Results, however, take a few minutes to a few hours, making these methods absolutely unsuitable for real-time applications, e.g. regulation of a rolling mill, are. In addition, of course, no dynamic influences can be taken into account as the solution is only for the calculation with the given boundary conditions Is valid.
Ziel dieser Methoden ist es, die Walzspaltkontur in Echtzeit zu berechnen. Da diese Berechnungen naturgemäß sehr zeitkritisch sind, können nur Näherungsverfahren angewandt werden. Dazu werden existierende Lösungen der Elastizitätstheorie, wie ein eingespannter Träger unter Volumenkraft, unter Querkraft bzw. Momentbelastung, oder die Deformation eines elastischen Halbraumes unter lokal wirkender Kraft, kombiniert, wodurch diese Methoden zwar sehr schnell arbeiten, aufgrund der Näherungsverfahren, durch die anhaftende Ungenauigkeit dieser Verfahren, jedoch nur eingeschränkt brauchbare bzw. sogar unbrauchbare Ergebnisse liefern.The aim of these methods is to calculate the roll gap contour in real time. This one Calculations are naturally very time-critical, only approximate methods can be applied. To this end, existing solutions to elasticity theory, such as a clamped beam under volume force, under shear force or moment load, or the deformation of an elastic half space under locally acting force, combined, which means that these methods work very quickly because of the approximation methods, due to the inherent inaccuracy of these methods, but only to a limited extent provide usable or even useless results.
Ein Verfahren zur Online-Berechnung der Walzspaltkontur geht z.B. aus der US 5,873,277 A hervor. Diesem Verfahren liegt aber ebenfalls eine Näherung zugrunde, indem die Walzen des Walzgerüsts für die Berechnung in mehrere Scheiben zerlegt werden und dazwischen interpoliert wird. Damit weist dieses Verfahren eine beschränkte Genauigkeit auf, da die Scheiben nicht beliebig dünn gemacht werden können, da ansonsten die Rechenzeit zu rasch ansteigt und sich das Verfahren folglich nicht mehr für eine Echtzeitanwendung eignen würde.A method for online calculation of the roll gap contour is e.g. from US 5,873,277 A out. However, this method is also based on an approximation in that the rollers of the roll stand can be broken down into several disks for the calculation and in between is interpolated. This method therefore has a limited accuracy because the Slices cannot be made arbitrarily thin, otherwise the computing time increases increases rapidly and the method is therefore no longer suitable for real-time application would.
Ziel der vorliegenden Erfindung ist es deshalb, ein Verfahren und eine Vorrichtung zur Berechnung der Walzspaltkontur anzugeben, dass sehr genaue Ergebnisse liefert und trotzdem schnell genug für Echtzeit-Anwendungen ausgeführt werden kann.The aim of the present invention is therefore to provide a method and an apparatus for Calculation of the roll gap contour to indicate that gives very accurate results and can still run fast enough for real-time applications.
Die Aufgabe wird erfindungsgemäß dadurch gelöst, dass die Walzspaltkontur online aus den Ergebnissen einer vorab durchgeführten Vorab-Berechnung der Walzspaltkontur und einer online durchgeführten Online-Berechnung der Walzspaltkontur zusammengesetzt wird. Diese Vorgangsweise ermöglicht es, die Vorteile einer Vorab-Berechnung, die hohe Genauigkeit, und einer Online-Berechnung, die große Geschwindigkeit, in einem Verfahren zu vereinen, wodurch es möglich wird, die Walzspaltkontur sehr schnell und mit hoher Genauigkeit zu berechnen. Dieses Verfahren ist deshalb einfach in eine Echtzeit-Anwendung, z.B. eine Regelung einer Walzstraße, einzubinden. Darüber hinaus kann durch die hohe Genauigkeit des Verfahrens die Qualität des Walzproduktes gesteigert werden, da vorgegebene Walzprofile sehr genau eingehalten werden können.The object is achieved in that the roll gap contour online from the Results of a preliminary calculation of the roll gap contour and a online calculation of the roll gap contour is put together. This procedure enables the advantages of a pre-calculation, the high accuracy, and an online calculation, the high speed, in one process unite, which makes it possible to form the roll gap contour very quickly and with high accuracy to calculate. This procedure is therefore simple in a real time application, e.g. control of a rolling mill. In addition, the high accuracy of the process, the quality of the rolled product can be increased, since predetermined Rolled profiles can be adhered to very precisely.
Besonders vorteilhaft ist es, wenn als Lösung der Vorab-Berechnung ein Deformationsvektorfeld, bzw. das radiale Deformationsfeld, einer Walze unter einer vorgegebenen Belastung berechnet wird. Die Lösung lässt sich dabei sehr einfach auffinden, wenn die Lösung als Fourierreihe dargestellt wird.It is particularly advantageous if, as a solution to the preliminary calculation, a deformation vector field, or the radial deformation field, a roller under a predetermined Load is calculated. The solution can be found very easily if the Solution is represented as a Fourier series.
Ein sehr günstiges Verfahren ergibt sich, wenn die Lösung der Vorab-Berechnung mit einer Finite-Element Berechnung berechnet wird, da diese Berechnungen sehr genau sind und somit eine genaue Lösung des Problems ermittelt wird. Für die Erzielung einer bestimmten Genauigkeit ist es ausreichend, für die Lösung die ersten NT Fouriermoden zu berechnen. Alternativ dazu kann die Lösung der Vorab-Berechnung als Summe einer Lösung einer Finite-Element Berechnung und einer Lösung einer semi-analytischen Berechnung berechnet wird. Die Lösung der semi-analytischen Berechnung kann einfach aufgefunden werden, wenn die Lösung für einen unendlich langen Zylinder berechnet wird. Für die Erzielung einer bestimmten Genauigkeit ist es hierbei vorteilhaft, mit der Finite-Elemente Berechnung die ersten NF Fouriermoden der Lösung und mit der semi-analytischen Berechnung die NF+1 bis NT Fouriermoden der Lösung zu berechnen.A very favorable method results if the solution of the preliminary calculation is calculated using a finite element calculation, since these calculations are very precise and thus an exact solution to the problem is determined. To achieve a certain accuracy, it is sufficient to calculate the first N T Fourier modes for the solution. Alternatively, the solution of the preliminary calculation can be calculated as the sum of a solution of a finite element calculation and a solution of a semi-analytical calculation. The solution of the semi-analytical calculation can easily be found if the solution is calculated for an infinitely long cylinder. In order to achieve a certain accuracy, it is advantageous here to calculate the first N F Fourier modes of the solution with the finite element calculation and the N F +1 to N T Fourier modes of the solution with the semi-analytical calculation.
Wenn die Lösung für eine Walze mit normiertem Radius und/oder unter normierter Belastung berechnet wird, ergibt sich ein besonders vorteilhaftes Verfahren, da dann bestimmte Berechnungen nur ein einziges Mal durchgeführt werden müssen. In der Online-Berechnung können die normierten Lösungen dann sehr schnell durch eine geeignete Transformation an die realen Gegebenheiten angepasst werden, was die erforderliche Berechnungszeit verringert.If the solution for a roller with a standardized radius and / or under standardized load is calculated, there is a particularly advantageous method, since then certain Calculations only need to be done once. In the online calculation can then adopt the standardized solutions very quickly by means of a suitable transformation the real conditions are adjusted, what is the required calculation time reduced.
Besonders vorteilhaft werden die vorab ermittelten Lösungen dazu verwendet, online das Kontaktproblem zwischen Arbeitswalze und Walzmaterial und gegebenenfalls das Kontaktproblem zwischen weiteren sich berührenden Walzen zu berechnen. Zur Lösung wird also auf die Ergebnisse einer während des Walzenwechsels durchgeführten Vorab-Berechnung zurückgegriffen. Diese Ergebnisse müssen bei Bedarf nur mehr aus einem Speicher ausgelesen werden, was die Berechnung der Walzspaltkontur sehr beschleunigt und in Echtzeit-Anwendungen anwendbar macht.The previously determined solutions are used particularly advantageously to do this online Contact problem between work roll and rolling material and possibly that To calculate the contact problem between other rollers in contact. The solution that is, the results of a preliminary calculation performed during the roll change resorted. These results only need to come from one Memory can be read out, which speeds up the calculation of the roll gap contour very much and makes it applicable in real-time applications.
Eine weitere Verbesserung des Verfahren ergibt sich durch eine geeigneten Transformation des zweidimensionalen Kontaktproblems auf ein eindimensionales Kontaktproblem, wobei mit der Lösung aus der Vorab-Berechnung die Walzspaltkontur online anhand des eindimensionalen Kontaktproblems zwischen sich berührenden Walzen und/oder zwischen der Arbeitswalze und dem Walzmaterial berechnet wird und die eindimensionale Lösung im Anschluss auf die zweidimensionale Lösung rücktransformiert wird.A further transformation results in a suitable transformation the two-dimensional contact problem to a one-dimensional contact problem, whereby with the solution from the preliminary calculation, the roll gap contour online using the one-dimensional contact problem between contacting rollers and / or between the work roll and the rolling material is calculated and the one-dimensional solution in Connection to the two-dimensional solution is transformed back.
Sehr vorteilhaft kann das nichtlineare Kontaktproblem durch Linearisierung iterativ gelöst werden.The nonlinear contact problem can be solved very advantageously iteratively by linearization become.
Durch Anwendung des erfindungsgemäßen Verfahrens kann darüber hinaus beim Kalibrieren eines Walzgerüstes die Walzendeformationen einer Anzahl w Walzen des Walzgerüstes direkt aus den sich ergebenden w-1 gekoppelten Kontaktproblemen berechnet werden.By using the method according to the invention, the Calibrate a roll stand the roll deformations of a number w rolls of Roll stand calculated directly from the resulting w-1 coupled contact problems become.
Um eine geforderte Banddicke zu erreichen, kann eine Korrektur zur Bandaustrittsdicke zumindest eines Walzgerüstes aus der Differenz der Walzendeformation beim Kalibrieren und beim herkömmlichen Walzvorgang in Echtzeit berechnet werden und die Bandaustrittsdicke in Echtzeit bei Bedarf durch Verändern von Stellgrößen korrigiert werden. Zusätzlich kann aus dem Vergleich der Berechnungen beim Kalibrieren und beim herkömmlichen Walzvorgang die gemessene Gerüstauffederungskennlinie im Arbeitspunkt korrigiert werden. Mit den verfahrensmäßig berechneten Banddicken kann eine einfache Kontrolle der Toleranzhaltigkeit durchgeführt werden. Da die Ergebnisse des Verfahrens sehr genau sind, kann die Qualität der Walzprodukte, durch verbesserte Toleranzhaltigkeit bzw. durch die Einhaltung engerer Toleranzen, verbessert werden, was sich in weiterer Folge natürlich auch wirtschaftlich positiv auswirkt.In order to achieve a required strip thickness, a correction to the strip exit thickness can be made at least one roll stand from the difference in roll deformation during calibration and can be calculated in real time in the conventional rolling process and the Strip exit thickness can be corrected in real time if necessary by changing manipulated variables. In addition, the comparison of the calculations when calibrating and when conventional rolling process the measured scaffold suspension characteristic at the working point Getting corrected. With the strip thicknesses calculated according to the process, a simple one Check the tolerance tolerance. Because the results of the procedure are very accurate, the quality of the rolled products, through improved tolerance or by adhering to tighter tolerances, which can be further improved Consequence of course also has a positive economic impact.
Sehr vorteilhaft wird das erfindungsgemäße Verfahren in einer übergeordneten Regelung einer Walzstraße eingebunden, die die Walzspaltkontur, und gegebenfalls die Bandaustrittsdicke, in Echtzeit berechnet, mit einem vorgegebenen Wert vergleicht und außerhalb der vorgegebenen Toleranz liegende Abweichungen der Walzspaltkontur bzw. der Banddicke in Echtzeit durch Verändern von Stellgrößen korrigiert. Damit hat man die Möglichkeit, das Walzprofil angefangen vom ersten bis zum letzten Walzgerüst genau zu steuern. Die Einstellungen der einzelnen Walzgerüste können auf einander abgestimmt werden und so die Qualität des Walzproduktes weiter verbessert werden.The method according to the invention is very advantageous in a higher-level control integrated a rolling mill, the roll gap contour, and possibly the Strip exit thickness, calculated in real time, compared with a specified value and deviations of the roll gap contour or the lying outside the specified tolerance Corrected strip thickness in real time by changing manipulated variables. So you have that Possibility to exactly close the rolled profile from the first to the last roll stand Taxes. The settings of the individual roll stands can be coordinated and the quality of the rolled product can be further improved.
Ganz besonders vorteilhaft wird das Verfahren auf einem Computer in Form eines Computerprogramms implementiert, da dann das Verfahren sehr einfach und sehr flexibel an sich ändernde Verhältnisse angepasst, bzw. sehr einfach erweitert werden kann.The method on a computer in the form of a Computer program implemented, because then the process is very simple and very flexible changing circumstances can be adapted or expanded very easily.
Das erfindungsgemäße Verfahren zur Berechnung der Walzspaltkontur wird anhand der
Figuren 1 bis 3 und der folgenden Beschreibung beispielhaft und nicht einschränkend
beschrieben. Dabei zeigt
Die folgenden Beschreibungen beziehen sich auf die Fig. 1 bis 3. Zuerst wird die
Deformation einer einzelnen Walze unter einer radialen Druckbelastung p(r,ϕ,z)
berechnet. Die zugrundeliegende Beziehung dazu, ist die sich aus der differentiellen
Gleichgewichtsbedingung, ∂ j σ ij + ρgi = 0, und dem verallgemeinerten Hook'schen Gesetz,
mit uij = ∂ jui +∂ iuj / 2, ergebende bekannte Lamé-Gleichung in
der Form:
r,ϕ,z bezeichnen die Zylinderkoordinaten in Radial-, Winkel- und Achsenrichtung, u das
Deformationsvektorfeld u(r,ϕ,z) mit Komponenten in Richtung r,ϕ,z, E den
Elastizitätsmodul, ν die Querdehnungszahl, ρ die Dichte des Walzenmateriales, g die
Gravitationsbeschleunigung, σij den Spannungstensor und uij den Verformungstensor.
Dieses gekoppelte System linearer partieller Differentialgleichungen ist unter Zuhilfenahme
von geeigneten Randbedingungen,
σrr = p(Rw,ϕ,z)
σrz = 0 bzw. σrϕ = 0 im Ballenbereich, mit dem Radius Rw der Walze in Walzenmitte, bzw.
σrr = p(RL,ϕ,z)
σrz = 0 bzw. σrϕ = 0 im Bereich der Walzenlager, mit dem Zapfenradius RL im Bereich des
Walzenlagers,
zu lösen. r , ϕ, z denote the cylindrical coordinates in the radial, angular and axial directions, u the deformation vector field u ( r , ϕ, z ) with components in the direction r , ϕ, z , E the modulus of elasticity, ν the transverse expansion factor, ρ the density of the Roll material, g the acceleration of gravity, σ ij the tension tensor and u ij the deformation tensor. This coupled system of linear partial differential equations is with the help of suitable boundary conditions,
σ rr = p (R w , ϕ, z)
σ rz = 0 or σ rϕ = 0 in the bale area , with the radius R w of the roll in the middle of the roll, or σ rr = p (R L , ϕ, z)
σ rz = 0 or σ rϕ = 0 in the area of the roller bearings , with the pin radius R L in the area of the roller bearings,
to solve.
Im Fall einer konischen Lagerform sind σrr, σrz, σrϕ durch σnn, σnt, σnϕ zu ersetzen, wobei σnn, σnt, σnϕ die Komponenten des Spannungstensors in transformierten Koordinaten sind, n bezeichnet die Normalenrichtung bzgl. des Lagerkonus, t ist die entsprechende Transversalrichtung.In the case of a conical bearing shape, σ rr , σ rz , σ rϕ are to be replaced by σ nn , σ nt , σ nϕ , where σ nn , σ nt , σ nϕ are the components of the stress tensor in transformed coordinates, n denotes the normal direction with respect to of the bearing cone, t is the corresponding transverse direction.
Aufgrund der Linearität der Lamé-Gleichung lässt sich jede Lösung zu einer gegebenen
Druckverteilung p(R,ϕ, z) als Superposition der Lösungen zu speziellen Druckverteilungen
pij (R,ϕ,z) darstellen, mit i = 1,...,Nz und j = 1,...,Nϕ:
LB bezeichnet hierbei die Ballenlänge der Walze. Wegen der Rotationssymmetrie genügt es allerdings, die Lösungen zu den Druckverteilungen pi 0(R,ϕ,z), i = 1,...,Nz, zu bestimmen.L B denotes the bale length of the roller. However, because of the rotational symmetry, it is sufficient to determine the solutions to the pressure distributions p i 0 ( R , ϕ, z ), i = 1, ..., N z .
Jede Lösung L der Lamé-Gleichung kann in allgemein bekannter Form als Fourierreihe
dargestellt werden. Zur Lösung wird der spezielle Ansatz L = LFEM + LANL gewählt.
L FEM / n(r,z) bezeichnet in Folge die n-te Fouriermode einer Lösung der Lamé-Gleichung,
berechnet mit der Methode der Finite-Elemente im Fourierraum und L ANL / n(r,z) bezeichnet die
n-te Fouriermode einer Lösung der Lamé-Gleichung, berechnet mit semi-analytischen
Methoden für einen unendlich langen Zylinder. Die gesamte Lösung L wird also aus einer
Finite-Elemente Lösung LFEM und einer semi-analytischen Lösung LANL konstruiert.
Die Randbedingung σrr = p(R,ϕ,z) ist in analoger Weise durch die fouriertransformierte Form
σrr n = pn(R,z) zu ersetzen.
Aufgrund der speziellen Eigenschaften der zylinderförmigen Walzengeometrie ist der
Gravitationsbeitrag sowie die Lagerkraft nur in der ersten Fouriermode (n=1) zu
berücksichtigen.Each solution L of the Lamé equation can be represented in a generally known form as a Fourier series. The special approach L = L FEM + L ANL is chosen for the solution. L FEM / n ( r , z ) consequently denotes the nth Fourier mode of a solution of the Lamé equation, calculated using the finite element method in the Fourier space, and L ANL / n ( r , z ) denotes the nth Fourier mode a solution of the Lamé equation, calculated using semi-analytical methods for an infinitely long cylinder. The entire solution L is thus constructed from a finite element solution L FEM and a semi-analytical solution L ANL . The boundary condition σ rr = p (R, ϕ, z) must be replaced in an analogous manner by the Fourier-transformed form σ rr n = p n (R, z).
Due to the special properties of the cylindrical roller geometry, the gravitational contribution and the bearing force must only be taken into account in the first Fourier mode (n = 1).
Es ist im Rahmen der Erfindung auch möglich, gänzlich auf semi-analytische Lösungen LANL zu verzichten und alle notwendigen Fouriermoden mit der Finite-Elemente-Berechnung zu ermitteln, d.h. L = LFEM .It is also possible within the scope of the invention to dispense entirely with semi-analytical solutions L ANL and to determine all necessary Fourier modes using the finite element calculation, ie L = L FEM .
Die Lösungen LFEM der Finite-Element Berechnung können mit den hinlänglich bekannten Methoden der Finiten-Elemente gefunden werden. Der beispielhafte Lösungsweg wird deshalb in Folge nur in groben Zügen skizziert. Die Lamé-Gleichung wird zuerst mit einer Testfunktion v multipliziert und anschließend über das Volumen V mit Oberfläche O aufintegriert. Mit der Darstellung eines Rotationskörpers K als Produkt von Rotationsfläche F und Winkelvariablen ϕ, K = F ⊗ ϕ, mit der Darstellung von F als Summe von Dreiecken Fd, und der Darstellung von u als Summe von geeigneten Testfunktionen u i / k cos(nϕ) bzw. u i / k sin(nϕ) mit Amplituden c i / kn und Integration über die Winkelvariable ϕ gelangt man in jedem Fouriermode zu einem linearen Gleichungssystem zur Bestimmung der Amplituden c i / kn. Für jeden der NF Fouriermoden sind also zu den Nz Druckverteilungen pi 0(R,ϕ,z), i = 1,...,Nz die entsprechenden Amplituden zu berechnen. Daraus folgen im Besonderen NF × Nz Lösungen für das radiale Deformationsfeld u FEM ,i / rn cos(nϕ), n = 0,...,NF-1 , i = 1,...,Nz.The solutions L FEM of the finite element calculation can be found using the well known methods of finite elements. The exemplary solution is therefore only outlined in broad outline. The Lamé equation is first multiplied by a test function v and then integrated over the volume V with surface O. With the representation of a body of revolution K as the product of the rotation surface F and the angle variable ϕ, K = F ⊗ ϕ, with the representation of F as the sum of triangles F d , and the representation of u as the sum of suitable test functions u i / k cos ( n ϕ) or u i / k sin ( n ϕ) with amplitudes c i / kn and integration via the angle variable ϕ leads to a linear in each Fourier mode System of equations for determining the amplitudes c i / kn . For each of the N F Fourier modes, the corresponding amplitudes are to be calculated for the N z pressure distributions p i 0 ( R , ϕ, z ), i = 1, ..., N z . In particular, this results in N F × N z solutions for the radial deformation field u FEM, i / rn cos ( n ϕ), n = 0, ..., N F -1, i = 1, ..., N z .
Zur Lösung des semi-analytischen Anteils der Gesamtlösung, wird der folgende Ansatz
gewählt:
Die Funktionen u r / n(r,z), u ϕ / n(r,z) und u z / n(r,z) werden selbst wieder als Fourierintegrale bzgl.
z dargestellt. Dies führt nach Einsetzen in die Lamé-Gleichung auf drei linear unabhängige
Sätze von Lösungen für den semi-analytischen Anteil. Die Randbedingung σrr n = pn(R,z)
wird, wie aus den obigen Ausführungen bereits bekannt, ebenfalls als Fourierintegral
dargestellt.
Durch Linearkombination der drei sich ergebenden Lösungssätze können diese
Randbedingungen erfüllt werden. Numerische Integration der so berechneten Lösungen
bzgl. k liefert dann die Lösung L ANL / n des Randwertproblems für die n-te Fouriermode.The functions u r / n ( r , z ), u ϕ / n ( r , z ) and u z / n ( r , z ) are again represented as Fourier integrals with respect to z. After insertion into the Lamé equation, this leads to three linearly independent sets of solutions for the semi-analytical part. The boundary condition σ rr n = p n (R, z) is, as already known from the above explanations, also represented as a Fourier integral.
These boundary conditions can be met by linear combination of the three resulting solution sets. The solution L ANL / n of the boundary value problem for the nth Fourier mode then provides numerical integration of the solutions calculated in this way with respect to k.
Zu beachten ist hier besonders, dass die Lösung für eine auf Radius R = 1 normierte Walze
ermittelt werden kann. Multiplikation des so berechneten Deformationsvektorfeldes mit dem
aktuellen Walzenradius liefert dann das aktuelle Deformationsvektorfeld. Es werden dabei im
Besonderen die speziellen Randbedingungen
Es wird also einmalig die normierte Lösung L ANL / c 0ϕ0 berechnet und erst bei Bedarf die
Transformation der Lösungen auf die tatsächliche Geometrie durchgeführt.
Es ist natürlich auch möglich, direkt für die jeweilige Geometrie der Walze diese Lösungen
L ANL / cϕ zu bestimmen. Die Berechnung dieser Lösungen L ANL / cϕ beansprucht jedoch sehr viel
Rechenzeit, weshalb es günstiger ist, die Berechnungen nur einmal durchzuführen und dann
bei Bedarf lediglich die Transformationen durchzuführen. It is particularly important to note that the solution can be determined for a roller standardized to radius R = 1. Multiplication of the deformation vector field calculated in this way by the current roller radius then delivers the current deformation vector field. In particular, it will be the special boundary conditions
The standardized solution L ANL / c 0 ϕ 0 is thus calculated once and the transformation of the solutions to the actual geometry is carried out only when required.
It is of course also possible to determine these solutions L ANL / c ϕ directly for the respective geometry of the roll. However, the calculation of these solutions L ANL / c ϕ takes up a lot of computing time, which is why it is cheaper to carry out the calculations only once and then only to carry out the transformations if necessary.
Auf diese Weise wird die Lösung L der Lamé-Gleichung ermittelt, wodurch der Deformationszustand
einer einzelnen Walze, hervorgerufen durch eine radiale Druckbelastung p(r, ϕ,z)
bekannt ist. Während des Walzbetriebes ergibt sich jedoch im Walzgerüst 1, wie in Fig. 1
bzw. Fig. 2 dargestellt, eine Berührung zwischen der Arbeitswalze A und dem Walzmaterial
M und zwischen der Arbeitswalze A und einer Stützwalze S. Bei mehrstöckigen
Walzgerüsten können sich auch noch zwei oder mehrere Stützwalzen berühren, wodurch
sich ein mehrfaches Kontaktproblem ergibt, das gelöst werden muss, um die aktuelle
Walzspaltkontur zu erhalten. Es wird im Folgenden beispielhaft der obere Walzensatz
hergenommen und die Lösung des Kontaktproblems anhand dieses Walzensatzes
beschrieben. Dieser Lösungsweg ist dann natürlich auch für den unteren und alle anderen
Walzensätze, auch für solche, wo nur Arbeitswalzen A vorhanden sind, äquivalent
anzuwenden.In this way, the solution L of the Lamé equation is determined, whereby the state of deformation of an individual roller, caused by a radial pressure load p ( r , ϕ , z ), is known. During the rolling operation, however, there is contact between the work roll A and the rolling material M and between the work roll A and a backup roll S in the
Während des Walzvorgangs, Fig. 2, ergibt sich für den oberen Walzensatz folgende
Situation: Die Arbeitswalze A ist an der Unterseite des Ballens über die Bandbreite B in
Kontakt mit dem Walzmaterial M und an der Oberseite in Kontakt mit der Stützwalze S. In
den Lagern der Arbeitswalze wirken Biegekräfte FB. Die Position der Arbeitswalze A kann
quer zur Walzrichtung eine Verschiebung dA aufweisen. Die Position des Walzmaterials M
kann ebenfalls quer zur Walzrichtung verschoben sein, Verschiebung dM. Zwischen
Walzmaterial M und Arbeitswalze A wirkt die Walzkraft FW.
Die Stützwalze S ist an ihrer Unterseite in Kontakt mit der Arbeitswalze A. In den Lagern der
Stützwalze S wirken die Ständerkräfte FS.
Sowohl Arbeits- A als auch Stützwalze S weisen einen Schliff auf (sA,sS), beide Walzen sind
thermisch gedehnt (tA,tS) und ihre Kontur ist durch Verschleiß verändert (vA,vS). Diese
Einflüsse sind als bekannt anzusehen und können entweder direkt aus Messungen bestimmt
werden, oder stammen wiederum aus geeigneten Modellrechnungen.
Die Kontaktfläche KAM zwischen Arbeitswalze A und Walzmaterial M, bezogen auf das
Koordinatensystem der Arbeitswalze A, wird folgendermaßen beschrieben:
During the rolling process, FIG. 2, the following situation arises for the upper set of rolls: the work roll A is in contact with the rolling material M on the underside of the bale over the strip width B and in contact with the backup roll S in the top. In the bearings bending forces F B act on the work roll. The position of the work roll A can have a displacement d A transverse to the rolling direction. The position of the rolling material M can also be shifted transversely to the rolling direction, shift d M. The rolling force F W acts between the rolling material M and the work roll A.
The support roller S is in contact with the work roller A on its underside. The stator forces F S act in the bearings of the support roller S.
Both work roller A and back-up roller S are ground (s A , s S ), both rollers are thermally expanded (t A , t S ) and their contour is changed by wear (v A , v S ). These influences can be regarded as known and can either be determined directly from measurements, or in turn come from suitable model calculations.
The contact area K AM between work roll A and rolling material M, based on the coordinate system of work roll A, is described as follows:
Der Kontaktwinkel Φk ergibt sich aus der bzgl. z maximalen Kontaktlänge Lk, dividiert durch
den Radius der Arbeitswalze A. Die Kontaktfläche KAM wird zerlegt in rechteckige Intervalle
Die Kontaktfläche KAS zwischen Arbeitswalze A und Stützwalze S, bezogen auf das Koordinatensystem der Arbeitswalze A, wird analog folgendermaßen beschrieben: The contact area K AS between work roll A and backup roll S, based on the coordinate system of work roll A, is described analogously as follows:
ΦS ist dabei wieder der bzgl. z maximale auftretende Kontaktwinkel und LBA bzw. LBS sind die
Ballenlängen von Arbeits- A und Stützwalze S. Die Kontaktfläche KAS wird wiederum zerlegt
in rechteckige Intervalle
Zu einer normierten radialen Druckverteilung P AM / ij auf die Kontaktfläche KAM in der Form
Die Indizes i und j beschreiben folglich den Angriffspunkt des Druckes p und die Indizes k
und I beschreiben den Ort wo die Deformation auftritt. Analog dazu ergibt sich zu einer normierten radialen Druckverteilung Q AS / ij auf die Kontaktfläche KAS in der Form
Bei gegebenen tatsächlichen Druckverteilungen p AM / ij auf KAM und q AS / ij auf KAS ergibt sich für die Gesamtdeformation an der Arbeitswalzenunterseite somit die Beziehung an der Arbeitswalzenoberseite die Beziehung und an der Stützwalzenunterseite die Beziehung Given the actual pressure distributions p AM / ij on K AM and q AS / ij on K AS , the relationship for the total deformation on the underside of the work roll is obtained the relationship on the top of the work roll and the relationship at the bottom of the backup roller
Dabei bezeichnen s u / AK, t u / AK, ν u / AK den Arbeitswalzenschliff, -dehnung und -verschleiß bezogen
auf die Diskretisierung zwischen Arbeitswalze A und Walzmaterial M, d.h. k = 1,...,N AM / z,
s o / AK,t o / AK,ν o / AK den Arbeitswalzenschliff, -dehnung und -verschleiß bezogen auf die
Diskretisierung zwischen Arbeitswalze A und Stützwalze S, d.h. k = 1,...,N AS / z und
sSK,tSK,νSK den Stützwalzenschliff, -dehnung und -verschleiß bezogen auf die
Diskretisierung zwischen Arbeitswalze A und Stützwalze S, d.h. k = 1,...,N AS / z. x0 und x1
beschreiben eine zusätzliche vertikale Verschiebung bzw. Verkippung der Arbeitswalze A.
Die Gesamtdeformation an der Arbeitswalzenunterseite ukl entspricht dabei genau der
gesuchten Walzspaltkontur, d.h. die Bestimmung der Walzspaltkontur ist gleichzusetzen mit
der Berechnung von ukl . Dazu wird das Kontaktproblem wie folgt formuliert:
rk ist dabei der nichtdeformierte" Abstand zwischen Arbeits- A und Stützwalze S. Zusätzlich
müssen Gesamtkraft und Gesamtmoment verschwinden, was zwei weitere Gleichungen zur
Bestimmung von x0 und x1 liefert.
Aus dem oben formulierten Kontaktproblem wird jetzt q AS / ij, x0 und x1 ermittelt. Zu beachten
ist dabei auch, dass wenn die Druckverteilung p AM / ij zwischen Arbeitswalze A und
Walzmaterial M als bekannt angenommen werden kann, q AS / ij, x0 und x1 direkt berechnet
werden können. Ist die Druckverteilung p AM / ij nicht bekannt, müssen p AM / ij, q AS / ij, x0 und x1
iterativ berechnet werden.
Mit den somit bestimmten q AS / ij, x0 und x1 bzw. p AM / ij kann nun die Walzspaltkontur, d.h. ukl ,
berechnet werden.r k is the non-deformed "distance between work roller A and support roller S. In addition, the total force and total moment must disappear, which provides two further equations for determining x 0 and x 1 .
Q AS / ij , x 0 and x 1 are now determined from the contact problem formulated above. It should also be noted that if the pressure distribution p AM / ij between work roll A and rolling material M can be assumed to be known, q AS / ij , x 0 and x 1 can be calculated directly. If the pressure distribution p AM / ij is not known, p AM / ij , q AS / ij , x 0 and x 1 must be calculated iteratively.
With the q AS / ij , x 0 and x 1 or p AM / ij thus determined, the roll gap contour, ie u kl , can now be calculated.
Das oben in Form von zwei Ungleichungen formulierte Kontaktproblem ist jedoch nichtlinear,
weshalb die Lösung auf iterativen Wege erfolgt. Der Ausgangspunkt für die iterative Lösung
kann beispielsweise ein liniearisiertes Gleichungssystem der Form
sein. Die Matrix M wird dabei durch die Komponenten oAS ij,kl und sAS ij,kl der Beziehungen für
okl und skl gebildet. Der Vektor b enthält dann folglich alle anderen Komponenten der
Beziehungen für okl und skl .
Eine deutliche Verminderung der Rechenzeit kann durch Reduktion des zweidimensionalen
Kontaktproblems auf ein eindimensionales, mittels einer geeigneten mathematischen
Transformation, erzielt werden. Das Kontaktproblem ist dabei nur entlang einer Linie
ϕ=konst. zu lösen und im Anschluss auf das zweidimensionale Problem umzurechnen.However, the contact problem formulated above in the form of two inequalities is non-linear, which is why the solution is done iteratively. The starting point for the iterative solution can be, for example, a linearized system of equations of the form his. The matrix M is formed by the components o AS ij, kl and s AS ij, kl of the relationships for o kl and s kl . The vector b then contains all the other components of the relationships for o kl and s kl .
A significant reduction in computing time can be achieved by reducing the two-dimensional contact problem to a one-dimensional one using a suitable mathematical transformation. The contact problem is only along a line ϕ = const. to solve and then convert to the two-dimensional problem.
Das obige Verfahren kann jedoch nicht nur zur Berechnung der Walzspaltkontur während
des Walzvorganges herangezogen werden, sondern es können auch die Walzendeformationen
beim Kalibrieren des Walzgerüstes 1 berechnet werden. Für Walzgerüste mit
einer Anzahl w Walzen ergeben sich dabei w-1 Kontaktprobleme, die gelöst werden müssen.
Beim Kalibrieren eines zweistöckigen Walzgerüstes 1, wie in Fig. 1 und 2 gezeigt, sind im
Gegensatz zum Walzvorgang die beiden Arbeitswalzen A in direkter Berührung. Daher sind
drei gekoppelte Kontaktprobleme zu lösen:
Zusätzlich müssen für beide Walzensätze separat und für das gesamte System jeweils Gesamtkraft und Gesamtmoment verschwinden, was die Variablen x o / 0,x o / 1,x m / 0,x m / 1,x u / 0,x u / 1 festlegt. Der Lösungsalgorithmus ist in weiterer Folge analog zu oben.In addition, the total force and total torque must disappear separately for both roller sets and for the entire system, which defines the variables x o / 0, x o / 1, x m / 0, x m / 1, x u / 0, x u / 1 , The solution algorithm is subsequently analogous to the one above.
Bei einer beispielhaften, praktischen Anwendung des erfindungsgemäßen Verfahrens in
Warmwalzwerken wird zuerst einmalig die Offline-Berechnung durchgeführt. Dabei wird die
normierte semi-analytische Lösung L ANL / c 0ϕ0 einmal berechnet (Rechenzeit ungefähr 20min;
Anm.: alle Angaben zur Rechenzeit sind beispielhaft und beziehen sich in Folge auf einen
PC mit 350MHz Taktfrequenz). Weiters werden, beispielsweise bei jedem Walzenwechsel,
als Vorab-Berechnung vorab die Fourier-Finite-Elemente-Lösungen und daraus die radialen
Deformationsfelder ermittelt (Rechenzeit ungefähr 40 sec pro Walze).
Im Betrieb werden danach bei Bedarf die Online-Berechnungen durchgeführt. Während der
nach dem Walzenwechsel notwendigen Kalibrierung der Walzgerüste 1 wird die dabei
auftretende Deformation der Walzen berechnet (Rechenzeit ungefähr 1sec pro Walzgerüst).
Alle diese Berechnungen müssen nur einmal durchgeführt werden. Während des
eigentlichen Walzvorganges wird dann bei Bedarf die aktuelle Walzspaltkontur, sowie jene
Korrektur zur Bandaustrittsdicke, die aus der Differenz der Walzendeformation beim
Kalibrieren und beim normalen Walzvorgang resultiert, berechnet. Zur präzisen Regelung
der Bandaustrittsdicke über die Bandlänge wird außerdem die genaue Steigung der
Gerüstauffederungskennlinie (= Änderung der Gerüstauffederung / Änderung der
Ständerkraft) im Arbeitspunkt benötigt. Diese folgt aus der gemessenen Kennlinie, korrigiert
um die berechnete Differenz der Steigungen beim Kalibrieren und beim normalen
Walzvorgang. (Rechenzeit ungefähr 0.05sec pro Walzensatz).
Alle rechenzeitintensiveren Berechnungen werden somit vorab bzw. offline durchgeführt. Die
eigentliche Online-Berechnung beansprucht nur sehr wenig Rechenzeit, ohne jedoch an
Genauigkeit einzubüßen, weshalb diese in Echtzeit durchgeführt werden kann. Dieses
Verfahren wird folglich in ein übergeordnetes Steuerungs- und Regelkonzept für die
Walzstraße eingebettet. Während des Walzvorgang kann für jedes einzelne Walzgerüst 1
jederzeit die Walzspaltkontur berechnet werden und mit vorgegebenen Werten verglichen
werden. Die Regelung kann bei Feststellen von Abweichungen durch Einflussnahme auf
gewisse Stellgrößen, wie beispielsweise die Biegekraft, die Walzenanstellung oder die
Arbeitswalzenverschiebung, die notwendigen Korrekturen vornehmen. In an exemplary, practical application of the method according to the invention in hot rolling mills, the offline calculation is first carried out once. The standardized semi-analytical solution L ANL / c 0 ϕ 0 is calculated once (computing time about 20min;
Note: all information on the computing time are exemplary and refer to a PC with a clock frequency of 350MHz). Furthermore, for example with each roll change, the Fourier finite element solutions and the radial deformation fields are determined in advance as a preliminary calculation (computing time approximately 40 seconds per roll).
The online calculations are then carried out in the company as required. During the calibration of the roll stands 1 which is necessary after the roll change, the deformation of the rolls which occurs is calculated (computing time approximately 1 second per roll stand). All of these calculations need only be done once. During the actual rolling process, the current roll gap contour and the correction to the strip exit thickness, which results from the difference in roll deformation during calibration and during the normal rolling process, are then calculated as required. For precise control of the strip exit thickness over the strip length, the exact slope of the scaffold spring characteristic (= change of the scaffold spring / change of the stator force) at the working point is also required. This follows from the measured characteristic curve, corrected by the calculated difference in the slopes during calibration and during the normal rolling process. (Computing time about 0.05sec per set of rollers).
All computation-intensive calculations are therefore carried out in advance or offline. The actual online calculation takes up very little computing time without losing accuracy, which is why it can be carried out in real time. This process is therefore embedded in a higher-level control and regulation concept for the rolling mill. During the rolling process, the roll gap contour can be calculated for each individual roll stand 1 at any time and compared with predetermined values. If deviations are determined by influencing certain manipulated variables, such as the bending force, the roll adjustment or the work roll displacement, the control can make the necessary corrections.
In Fig. 3 sind beispielhaft die Ergebnisse einer solchen Walzspaltkonturberechnung graphisch dargestellt. Im oberen Bild berühren sich die Stützwalzenunterseite und die Arbeitswalzenoberseite während des Walzvorganges. Die daraus resultierende Druckverteilung qAS ist in der mittleren Abbildung dargestellt. Man erkennt, dass an der Walze nur dort eine Druckbelastung auftritt, wo sich die beiden Walzen berühren, womit die Randbedingungen erfüllt sind. In der unteren Abbildung ist die berechnete Walzspaltkontur ukl dargestellt. Die kubische Form der Walzspaltkontur ukl ergibt sich dabei aus dem angewendeten kubischen Arbeitswalzenschliff.The results of such a roll gap contour calculation are shown graphically in FIG. 3. In the picture above, the bottom of the backup roll and the top of the work roll touch during the rolling process. The resulting pressure distribution q AS is shown in the middle figure. It can be seen that the roller is only subjected to pressure where the two rollers touch, which means that the boundary conditions are met. The calculated roll gap contour u kl is shown in the figure below. The cubic shape of the roll gap contour u kl results from the cubic work roll grinding used.
Claims (26)
- Method for calculating the roll gap contour in a roll stand, consisting of two working rolls (A), characterised in that the roll gap contour is composed online from the results of a prior calculation of the roll gap contour carried out beforehand, and an online calculation of the roll gap contour carried out online.
- Method according to Claim 1, characterised in that a deformation vector field u(r,ϕ,z), or the radial deformation field u r, of a roll under a predetermined load is calculated as the solution L of the prior calculation.
- Method according to Claim 2, characterised in that the solution L is represented as a Fourier series.
- Method according to Claim 3, characterised in that the solution L of the prior calculation is calculated by a finite-element calculation LFEM.
- Method according to Claim 4, characterised in that the first NT Fourier modes are calculated for the solution L.
- Method according to Claim 3, characterised in that the solution L of the prior calculation is calculated as the sum of a solution of a finite-element calculation LFEM and a solution of a semi-analytical calculation LANL.
- Method according to Claim 6, characterised in that the solution of the semi-analytical calculation LANL is calculated as a solution for an infinitely long cylinder.
- Method according to Claim 6 or 7, characterised in that the first NF Fourier modes of the solution L are calculated by the finite-element calculation, and the NF+1 to NT Fourier modes of the solution L are calculated by the semi-analytical calculation.
- Method according to one of Claims 1 to 8, characterised in that the solution L is calculated for a roll with a normalised radius and/or under a normalised load.
- Method according to Claim 9, characterised in that the normalised solutions are matched to the real situation by a suitable transformation in the online calculation and/or the prior calculation.
- Method according to one of Claims 1 to 10, characterised in that the roll gap contour is calculated online as the online solution with the solution from the prior calculation, from the contact problem at least between the working rolls (A) and the material (M) to be rolled and optionally between other rolls which touch.
- Method according to one of Claims 1 to 11, characterised in that the two-dimensional contact problem is reduced to a one-dimensional contact problem by a suitable transformation, the roll gap contour is calculated online with the solution from the prior calculation with the aid of the one-dimensional contact problem between rolls which touch and/or between the working rolls (A) and the material (M) to be rolled, and the one-dimensional solution is subsequently transformed back to the two-dimensional solution.
- Method according to Claim 11 or 12, characterised in that the nonlinear contact problem is solved iteratively.
- Method according to Claim 11 or 12, characterised in that the nonlinear contact problem is converted into a linear equation system by linearisation and solved iteratively.
- Method according to one of Claims 11 to 14, characterised in that the roll deformations of a number w of rolls of the roll stand (1) are calculated from the resulting w - 1 coupled contact problems when calibrating a roll stand (1).
- Method according to Claim 15, characterised in that a correction for the strip exit thickness of at least one roll stand (1) is calculated in real-time from the difference in roll deformation during the calibration and/or during the normal rolling process, and the strip exit thickness is corrected in real-time if required by varying setpoint quantities, for example the roll application setting.
- Method according to Claim 15 or 16, characterised in that the measured characteristic stand springing curve is corrected at the working point from comparison of the calculations during the calibration and during the normal rolling process.
- Method for the regulation of a rolling train with at least one roll stand, in which the roll gap contour of a roll stand (1) is calculated in real-time, compared with a predetermined value, and deviations of the roll gap contour lying outside the predetermined tolerance are corrected in real-time by varying setpoint quantities, for example the bending force and/or the working roll displacement, characterised in that the roll gap contour is calculated according to one of Claims 1 to 17.
- Method according to Claim 18, characterised in that the strip exit thickness of a roll stand (1) is calculated in real-time and the strip exit thickness is corrected in real-time by varying setpoint quantities, for example the roll application setting.
- Method according to Claim 18 or 19, characterised in that the measured characteristic stand springing curve can be corrected at the working point from comparison of the calculations during the calibration and during the normal rolling process.
- Device for calculating the roll gap contour in a roll stand, consisting of two working rolls and a calculation unit, characterised in that the roll gap contour can be calculated online in the calculation unit from the results of a prior calculation of the roll gap contour carried out beforehand, and an online calculation of the roll gap contour carried out online.
- Device according to Claim 21, characterised in that input quantities for the calculation of the roll gap contour, for example roll curvature, roll wear, roll displacements, forces, etc., can be measured and/or calculated from suitable models, and can be processed in the calculation unit.
- Device according to Claim 21 or 22, characterised in that the roll deformation can be calculated in real-time in the calculation unit during the calibration of a roll stand and/or during the rolling process, and the strip exit thickness can be corrected in real-time if required by varying setpoint quantities, for example the roll application setting.
- Device according to one of Claims 21 to 23, characterised in that the measured characteristic stand springing curve can be corrected at the working point from comparison of the calculations during the calibration and during the normal rolling process.
- Device according to one of Claims 21 to 24, characterised in that the calculation unit is a computer.
- Device according to Claim 25, characterised in that the calculation of the roll gap contour and/or roll deformation is implemented as a computer program on the computer.
Applications Claiming Priority (2)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
AT4002001A AT410904B (en) | 2001-03-14 | 2001-03-14 | METHOD AND DEVICE FOR CALCULATING THE ROLLER CONTOUR |
AT4002001 | 2001-05-17 |
Publications (2)
Publication Number | Publication Date |
---|---|
EP1240955A1 EP1240955A1 (en) | 2002-09-18 |
EP1240955B1 true EP1240955B1 (en) | 2004-07-14 |
Family
ID=3673545
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
EP20020001665 Expired - Lifetime EP1240955B1 (en) | 2001-03-14 | 2002-01-24 | Method and apparatus for calculating the roll gap contour |
Country Status (3)
Country | Link |
---|---|
EP (1) | EP1240955B1 (en) |
AT (1) | AT410904B (en) |
DE (1) | DE50200608D1 (en) |
Cited By (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN111801173A (en) * | 2018-03-09 | 2020-10-20 | 首要金属科技德国有限责任公司 | Edge wear prevention during rolling of flat rolling stock |
Families Citing this family (5)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
AT500764A1 (en) * | 2004-05-19 | 2006-03-15 | Voest Alpine Ind Anlagen | METHOD FOR CALCULATING THE GEOMETRIC FORM OF ROLLING MATERIAL |
DE102009043401A1 (en) * | 2009-09-29 | 2011-04-07 | Siemens Aktiengesellschaft | Method for the model-based determination of actuator setpoints for the symmetrical and asymmetric actuators of the rolling mills of a hot strip mill |
DE102017221126A1 (en) | 2017-11-27 | 2019-05-29 | Sms Group Gmbh | rolling mill |
CN110064670B (en) * | 2019-03-28 | 2020-08-18 | 南京钢铁股份有限公司 | Method for quickly calibrating pinch roll |
CN112559950B (en) * | 2020-11-16 | 2022-07-19 | 燕山大学 | Twenty-high rolling mill roll system elastic deformation roll unit dividing method for forecasting pressing phenomenon of edge part of working roll |
Family Cites Families (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
DE3637043A1 (en) * | 1986-10-30 | 1988-05-05 | Licentia Gmbh | Method for the predetermined maintenance of narrow thickness tolerances during the rolling of rolling stock in hot-strip rolling trains |
DE3829441A1 (en) * | 1988-07-11 | 1990-01-18 | Escher Wyss Ag | Apparatus for the cold rolling of metal strip and foil and a method for its operation |
DE19618712B4 (en) * | 1996-05-09 | 2005-07-07 | Siemens Ag | Control method for a roll stand for rolling a strip |
-
2001
- 2001-03-14 AT AT4002001A patent/AT410904B/en not_active IP Right Cessation
-
2002
- 2002-01-24 EP EP20020001665 patent/EP1240955B1/en not_active Expired - Lifetime
- 2002-01-24 DE DE50200608T patent/DE50200608D1/en not_active Expired - Lifetime
Cited By (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN111801173A (en) * | 2018-03-09 | 2020-10-20 | 首要金属科技德国有限责任公司 | Edge wear prevention during rolling of flat rolling stock |
CN111801173B (en) * | 2018-03-09 | 2022-06-14 | 首要金属科技德国有限责任公司 | Edge wear prevention during rolling of flat rolling stock |
Also Published As
Publication number | Publication date |
---|---|
EP1240955A1 (en) | 2002-09-18 |
AT410904B (en) | 2003-08-25 |
DE50200608D1 (en) | 2004-08-19 |
ATA4002001A (en) | 2003-01-15 |
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