Control method for inlet vertical roll lubricating device of hot continuous rolling finishing mill

Abstract

The invention discloses a control method for an inlet vertical roll lubricating device of a hot continuous rolling finishing mill. The control method comprises the following steps that S1, a vertical roll rolling force feedback control model is established; s2, establishing a rolling speed feedback control model; s3, establishing a strip steel billet center line deviation feedforward control model; s4, establishing a strip steel temperature characteristic compensation coefficient; and S5, the injection time sequence of the vertical roll lubricating device is set. The invention aims to solve the problems of poor lubricating effect, insufficient lubrication and excessive lubrication of the finish rolling vertical roll and improve the edge quality of a hot rolling strip steel finished product (especially high-grade silicon steel).

Description

Technical Field

[0001] This invention relates to hot rolling electrical technology, and more specifically, to a control method for the inlet vertical roll lubrication device of a hot continuous rolling mill finishing mill. Background Technology

[0002] Hot rolling lubrication technology has a long history of application in hot strip mills. It not only reduces energy consumption, increases productivity, lowers roll costs, and improves strip surface quality, but also improves the grain structure of the strip, resulting in ideal deep-drawing performance. With the expansion of product varieties, higher customer requirements, the development of short-process continuous casting and rolling processes, and the increasing market demand for hot-rolled strip steel with a thickness of less than 1 mm, hot rolling lubrication technology has attracted widespread attention and is now widely used. Currently, with the use of high-speed steel rolls, hot rolling lubrication technology is even more important. It involves mixing lubricating oil and cooling water in a specific ratio and then spraying the mixture onto the rolls under pressure. This forms a lubricating film on the contact surface between the work rolls and the workpiece, providing lubrication, reducing rolling force on the stand, and mitigating roll wear. Practice has proven that, under the same conditions, hot rolling lubrication is the most effective means of reducing rolling force. The width accuracy of hot-rolled strip is a crucial indicator of the quality of hot-rolled finished products. Precise width control means a stable and smaller width tolerance range, which not only improves product yield but also creates better production conditions for hot-rolling users and subsequent processes. The vertical rolls of the finishing mill are essential equipment for controlling the strip width in the finishing rolling area. In recent years, with downstream customers demanding increasingly higher product quality from hot-rolled production lines, improving strip edge quality and reducing edge defects caused by poor vertical roll surface conditions has become increasingly urgent. Currently, installing a finishing mill vertical roll lubrication device on the production line can improve strip edge quality, increase yield, and extend the vertical roll changeover cycle.

[0003] However, the following problems exist during use:

[0004] 1. In actual application, if the side pressure of the finishing vertical roll mill increases, the friction between the strip edge and the roll will increase. If the lubrication control is based on a fixed oil-water mixing ratio, the expected lubrication effect will not be achieved, the mill stand rolling force will be too large, and the roll wear will be aggravated.

[0005] 2. If the side pressure of the finishing vertical roll mill is reduced, the friction between the strip edge and the roll will decrease. If a fixed oil-water mixing ratio is used for lubrication control, the fixed oil-water mixing ratio will produce an excessive lubrication effect, which will easily cause the stand to slip.

[0006] 3. If the rolling speed of the finishing vertical roll mill increases, using a fixed oil-water mixing ratio will not achieve the expected lubrication effect, and the wear of the rolls will be aggravated.

[0007] 4. If the rolling speed of the finishing vertical roll mill decreases, using a fixed oil-water mixing ratio will result in excessive lubrication. Even when the rolled piece leaves the stand, there will still be lubricating oil adhering to the surface of the roll, which may cause slippage when the next strip is bitten. Summary of the Invention

[0008] To address the shortcomings of existing technologies, the present invention aims to provide a control method for the lubrication device of the inlet vertical roll of a hot strip mill, in order to solve the problem of poor lubrication effect of the vertical roll of the finishing mill, which often results in under-lubrication and over-lubrication, and to improve the edge quality of hot-rolled strip steel products (especially high-grade silicon steel).

[0009] To achieve the above objectives, the present invention adopts the following technical solution:

[0010] A method for controlling the lubrication device of the inlet vertical roll of a hot continuous rolling mill includes the following steps:

[0011] S1, Establish a vertical roll rolling force feedback control model;

[0012] S2, Establish a rolling speed feedback control model;

[0013] S3, Establish a feedforward control model for the deviation of the centerline of the strip billet;

[0014] S4, Establish the temperature characteristic compensation coefficient of strip steel;

[0015] S5 sets the spraying sequence of the vertical roller lubrication device.

[0016] Preferably, in step S1, the calculation formula for the vertical roll rolling force feedback control model is:

[0017]

[0018] In the formula, P1 n P1 represents the current output coefficient of the vertical roll rolling force feedback control model; n-1 PF represents the output coefficient of the vertical roll rolling force feedback control model at the previous moment. n F is the actual rolling force of the vertical roll at the current moment; F is the actual rolling force locked at the moment the vertical roll bites the steel; PF n-1 This represents the actual rolling force of the vertical roll at the previous moment.

[0019] Preferably, in step S2, the calculation formula for the rolling speed feedback control model is:

[0020]

[0021] In the formula, P2 n P2 represents the current output coefficient of the rolling speed feedback control model; n-1PS represents the output coefficients of the rolling speed feedback control model at the previous moment. n S is the actual rolling force of the vertical roll at the current moment; S is the actual rolling force locked at the moment the vertical roll bites the steel; PS n-1 This represents the actual rolling force of the vertical roll at the previous moment.

[0022] Preferably, in step S3, the feedforward control model for the deviation of the strip billet centerline includes a feedforward control model for the deviation of the strip billet centerline on the vertical roll drive side and a feedforward control model for the deviation of the strip billet centerline on the vertical roll working side.

[0023] Preferably, the calculation formula for the feedforward control model of the deviation of the billet centerline on the vertical roller drive side is:

[0024]

[0025] In the formula, P3 n The current-moment output coefficients of the feedforward control model for the deviation of the billet centerline on the vertical roller drive side belt; P3 n-1 The output coefficient of the feedforward control model for the deviation of the steel billet centerline on the vertical roller drive side belt at the previous moment; PW n PW represents the width deviation value collected by the width measuring instrument at the R2 exit point; W is the target value set for the R2 exit width of the strip billet; n-1 This represents the width deviation value at the previous acquisition time of the width measuring instrument at the R2 exit.

[0026] Preferably, the calculation formula for the feedforward control model of the deviation of the steel billet centerline on the working side of the vertical roll is:

[0027]

[0028] In the formula, P4 n The current-moment output coefficients of the feedforward control model for the deviation of the steel billet centerline on the working side of the vertical roll; P4 n-1 The previous moment's output coefficients of the feedforward control model for the deviation of the steel billet centerline on the working side of the vertical roll; PW n PW represents the width deviation value collected by the width measuring instrument at the R2 exit point; W is the target value set for the R2 exit width of the strip billet; n-1 This represents the width deviation value at the previous acquisition time of the width measuring instrument at the R2 exit.

[0029] Preferably, in step S4, the strip temperature characteristic compensation coefficient is as follows:

[0030] K10 = A10i * R10 + B10i

[0031] K11=A11i*R11+B11i

[0032] K12=A12i*R12+B12i

[0033] K13 = A13i * R13 + B13i

[0034] …

[0035] K19 = A19i * R19 + B19i

[0036] In the above formula, K1x is the strip edge temperature drop time characteristic layering compensation coefficient; A10~A19 are the strip edge temperature drop time characteristic compensation layering proportional gain coefficients (0~9); R10~R19 are the strip edge temperature ten-segment layering interval values; B10~B19 are the strip edge temperature drop time characteristic compensation layering adjustment coefficients (0~9); i is the temperature difference input; K is the output value corresponding to the temperature difference input.

[0037] Preferably, in step S5, the spraying timing of the vertical roller lubrication device is controlled as follows:

[0038] The spray timing T-open of the vertical roll lubrication device is set to be activated after the vertical roll bite signal is established plus a delay, that is:

[0039] T-open:F1E LOAD ON+TD

[0040] When the distance S0 between the tail of the rolled piece and the vertical roll is less than S1+α, the vertical roll lubrication spray device is turned off, that is:

[0041] T-close:S0 <S1+α

[0042] Where S1 is the distance of one rotation of the vertical roller, S1=π×R, R is the diameter of the vertical roller; α is the compensation value.

[0043] The present invention provides a control method for the lubrication device of the inlet vertical roll of a hot strip mill finishing mill. This method primarily changes the current lubrication control method for the vertical roll of the finishing mill by establishing a rolling force feedback control model, a speed feedback control model, and related timing control. It has the following beneficial effects:

[0044] 1. By improving the lubrication setpoint control method using existing equipment, the lubrication effect of the vertical rolls of the hot rolling finishing mill was improved, thereby improving the edge quality of hot-rolled products;

[0045] 2. When the rolling load of the finishing mill vertical roll is less than the traditional empirical value, the amount of lubricant used is reduced while achieving the lubrication effect;

[0046] 3. When the rolling load of the finishing mill vertical roll is greater than the traditional empirical value, it can ensure that the finishing mill vertical roll receives sufficient and effective lubrication;

[0047] 4. It has good versatility and high adaptability, and high reliability. It is suitable for other hot rolling mill vertical roll lubrication systems controlled by PLC logic programs. Attached Figure Description

[0048] Figure 1 This is a schematic diagram of the layout of the entrance vertical rolls and related equipment of the existing finishing mill;

[0049] Figure 2 This is a schematic diagram of the structural framework of the existing finishing mill inlet vertical roll lubrication control system;

[0050] Figure 3 This is a logical schematic diagram of the control method for the inlet vertical roller lubrication device of the present invention;

[0051] Figure 4 This is a logical schematic diagram of the vertical roll rolling force feedback control model in the inlet vertical roll lubrication device control method of the present invention;

[0052] Figure 5 This is a logical schematic diagram of the rolling speed feedback control model in the inlet vertical roll lubrication device control method of the present invention;

[0053] Figure 6 This is a schematic diagram showing the relationship between the R2 roughing mill and the F1E finishing mill vertical roll in the inlet vertical roll lubrication device control method of the present invention;

[0054] Figure 7 This is a screenshot of the data tracking curves of the mill rolling speed, rolling force feedback, and control output pump speed change lubrication after the vertical roll lubrication control is started in an embodiment of the vertical roll lubrication device control method of the present invention. Detailed Implementation

[0055] To better understand the above-mentioned technical solutions of the present invention, the technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0056] Combination Figure 1 As shown, a hot metal detector 2 is installed on the inlet side of the vertical roll 1 of the finishing mill (in the direction of the incoming billet material) to detect the position of the incoming material head. A finishing mill descaling box 3 and a descaling roller conveyor 4 are installed in front of the hot metal detector 2. The finishing mill F1 mill 5 is located on the outlet side of the vertical roll 1 of the finishing mill.

[0057] Combination Figure 2As shown, the finishing mill inlet vertical roll lubrication control system consists of a finishing mill inlet F1E control PLC, an F1E vertical roll lubrication control PLC, and frequency converters for three F1E lubrication pump motors. Specifically, lubrication pump #1 is responsible for lubricating the F1E vertical roll drive side; lubrication pump #2 is responsible for lubricating the F1E vertical roll working side; and lubrication pump #3 is a standby pump. The finishing mill inlet F1E control PLC is responsible for collecting the fixed lubricating oil-to-water ratio issued by L2, collecting feedback on the F1E vertical roll rolling force and mill speed, and calculating the lubrication start and end times. The F1E vertical roll lubrication control PLC receives relevant signals and values ​​from the finishing mill inlet F1E control PLC and controls the frequency converters to adjust the pump motor speed, thereby changing the oil-to-water ratio at the lubrication pipeline outlet to achieve better lubrication.

[0058] Combination Figure 3 As shown, the present invention provides a method for controlling the lubrication device of the inlet vertical roll of a hot continuous rolling mill, comprising the following steps:

[0059] S1, Establish a vertical roll rolling force feedback control model;

[0060] S2, Establish a rolling speed feedback control model;

[0061] S3, Establish a feedforward control model for the deviation of the centerline of the strip billet;

[0062] S4, Establish the temperature characteristic compensation coefficient of strip steel;

[0063] S5 sets the spraying sequence of the vertical roller lubrication device.

[0064] The calculation formula for the vertical roll rolling force feedback control model is:

[0065]

[0066] In the formula, P1 n P1 represents the output coefficients of the vertical roll rolling force feedback control model at the current time (time n); n-1 PF represents the output coefficient of the vertical roll rolling force feedback control model at the previous time step (time step n-1); n F is the actual rolling force of the vertical roll at the current moment (time n); F is the actual rolling force locked at the moment the vertical roll bites the steel; PF n-1 This represents the actual rolling force of the vertical roll at the previous moment (moment n-1).

[0067] Combination Figure 4As shown, the vertical roll rolling force feedback control model extracts the current rolling force F1E from the rolling force locked when F1E bites the steel, divides it, and then inputs it into the P controller as the proportional coefficient. Simultaneously, it compares the current rolling force F1E with the previous rolling force F1E as the input value to the P controller. This input value is then integrated by the I controller, added to the original P controller output value, and then input to the L controller. Finally, after amplitude limiting, the current (n) output coefficient P1n of the vertical roll rolling force feedback control model is obtained.

[0068] The calculation formula for the rolling speed feedback control model is:

[0069]

[0070] In the formula, P2 n P2 represents the output coefficients of the rolling speed feedback control model at the current time (time n); n-1 PS represents the output coefficients of the rolling speed feedback control model at the previous time step (time step n-1); n S is the actual rolling force of the vertical roll at the current moment (time n); S is the actual rolling force locked at the moment the vertical roll bites the steel; PS n-1 This represents the actual rolling force of the vertical roll at the previous moment (moment n-1).

[0071] Combination Figure 5 As shown, the rolling speed feedback control model extracts the current rolling force of the F1E vertical roll and divides it by the rolling speed locked when the F1E vertical roll bites the steel. This division is then fed into the P controller as the proportional coefficient. Simultaneously, the current rolling speed of the F1E vertical roll is compared with the previous rolling speed, and this comparison is used as the input value to the P controller. This input value is then integrated by the I controller and added to the original P controller output value before being fed into the L controller. Finally, after amplitude limiting processing, the current time (n) output coefficient P2n of the vertical roll lubrication rolling speed feedback control model is obtained.

[0072] The feedforward control model for the deviation of the centerline of the strip billet includes a feedforward control model for the deviation of the centerline of the strip billet on the vertical roll drive side and a feedforward control model for the deviation of the centerline of the strip billet on the vertical roll working side.

[0073] The calculation formula for the feedforward control model of the deviation of the center line of the steel billet on the vertical roller drive side is as follows:

[0074]

[0075] In the formula, P3 n The current-moment output coefficients of the feedforward control model for the deviation of the billet centerline on the vertical roller drive side belt; P3 n-1 The output coefficient of the feedforward control model for the deviation of the steel billet centerline on the vertical roller drive side belt at the previous moment; PW nPW represents the width deviation value collected by the width measuring instrument at the R2 exit point; W is the target value set for the R2 exit width of the strip billet; n-1 This represents the width deviation value at the previous acquisition time of the width measuring instrument at the R2 exit.

[0076] The calculation formula for the feedforward control model of the deviation of the center line of the strip steel billet on the working side of the vertical roll is as follows:

[0077]

[0078] In the formula, P4 n The current-moment output coefficients of the feedforward control model for the deviation of the steel billet centerline on the working side of the vertical roll; P4 n-1 The previous moment's output coefficients of the feedforward control model for the deviation of the steel billet centerline on the working side of the vertical roll; PW n PW represents the width deviation value collected by the width measuring instrument at the R2 exit point; W is the target value set for the R2 exit width of the strip billet; n-1 This represents the width deviation value at the previous acquisition time of the width measuring instrument at the R2 exit.

[0079] Combination Figure 6 As shown, the strip billet is first rolled by the R2 roughing mill, and then transported to the finishing mill area for rolling via the intermediate roller table. During the R2 rolling process, due to temperature changes on both sides of the strip, early side-pressure rolling deviations, and stiffness deviations on both sides of the roughing mill, the centerline deviation of the strip billet width in the cross-sectional direction will occur. This results in a larger contact area between the intermediate strip billet and one side of the F1E vertical roll for a period of time, increasing the friction and thus increasing roll surface wear; at the same time, the contact with the other side is correspondingly reduced, resulting in less friction and mitigating roll surface wear. Therefore, a feedforward control of the strip billet centerline deviation is adopted to address this phenomenon. By collecting the numerical values ​​of the centerline deviation of the R2 mill exit width, a feedforward compensation coefficient is calculated and used in the F1E lubrication control.

[0080] The establishment of the strip temperature characteristic compensation coefficient is as follows:

[0081] K10 = A10i * R10 + B10i

[0082] K11=A11 i*R11+B11 i

[0083] K12=A12i*R12+B12i

[0084] K13 = A13i * R13 + B13i

[0085] …

[0086] K19 = A19i * R19 + B19i

[0087] In the above formula, K1x (x = 0 to 9) is the stratification compensation coefficient for the temperature drop time characteristic of the strip edge; A10 to A19 are the proportional gain coefficients for the stratification compensation of the temperature drop time characteristic of the strip edge (0 to 9); R10 to R19 are the stratification interval values ​​for the temperature drop time characteristic of the strip edge (10 segments); and B10 to B19 are the adjustment coefficients for the stratification compensation of the temperature drop time characteristic of the strip edge (0 to 9).

[0088] Because the temperature of the strip decreases along its length, especially at the tail end where the temperature drop can reach 50 degrees Celsius or more, it is generally believed that the higher the temperature, the faster the lubricating oil burns, and the lower the temperature, the slower the lubricating oil burns. Therefore, a strip temperature characteristic compensation coefficient is established to optimize the lubricating oil injection quantity.

[0089] The specific control of the spray timing of the vertical roller lubrication device is as follows:

[0090] To ensure rolling stability and prevent slippage of the workpiece head during vertical roll rolling, the spray timing T-open of the vertical roll lubrication device is set to activate after a delay following the establishment of the vertical roll bite signal.

[0091] T-open:F1E LOAD ON+TD

[0092] To ensure that no lubricating oil residue remains on the vertical roll surface after the workpiece is rolled, thus preventing slippage during the rolling of the next workpiece, it is necessary to track the tail of the workpiece. Furthermore, the distance S1 of one rotation of the vertical roll needs to be calculated. When the distance S0 between the tail of the workpiece and the vertical roll is less than S1+α, the vertical roll lubrication spray device is turned off.

[0093] T-close:S0 <S1+α

[0094] Where S1 is the distance of one rotation of the vertical roller, S1=π×R, R is the diameter of the vertical roller; α is the compensation value.

[0095] This invention effectively solves the problem of poor lubrication of vertical rolls in finishing mills, frequently resulting in under-lubrication or over-lubrication, thus improving the edge quality of hot-rolled strip steel products (especially high-grade silicon steel). Furthermore, it is simple to implement, highly reliable, and can be widely applied to PLC-controlled vertical roll lubrication systems. Its feasibility and prospects for widespread application are very promising.

[0096] Example

[0097] This embodiment provides a method for controlling the lubrication device of the inlet vertical roll of a hot continuous rolling mill, including the following steps:

[0098] S1, Establish a vertical roll rolling force feedback control model;

[0099] S2, Establish a rolling speed feedback control model;

[0100] S3, Establish a feedforward control model for the deviation of the centerline of the strip billet;

[0101] S4, Establish the temperature characteristic compensation coefficient of strip steel;

[0102] S5 sets the spraying sequence of the vertical roller lubrication device.

[0103] like Figure 7 As shown, in this embodiment, during the control process, at time point X1, the rolling force feedback control compensation coefficient is 0.01; the speed feedback compensation coefficient is -0.0373; the strip temperature characteristic compensation coefficient is 9; and the total compensation coefficient is 1.8559.

[0104] At time point X2, the rolling force feedback control compensation coefficient is 0.02; the speed feedback compensation coefficient is 0.2599; the strip temperature characteristic compensation coefficient is 5.01; and the total compensation coefficient is 1.9172.

[0105] Those skilled in the art should recognize that the above embodiments are merely illustrative of the present invention and are not intended to limit the present invention. Any variations or modifications to the above embodiments that are within the spirit and essence of the present invention will fall within the scope of the claims of the present invention.

Claims

1. A control method for a lubricating device of an entry vertical roll of a hot continuous rolling finishing mill, characterized by, The method comprises the following steps: S1, establishing a vertical roll rolling force feedback control model; S2, establishing a rolling speed feedback control model; S3, establishing a strip blank center line deviation feedforward control model; S4, establishing a strip temperature characteristic compensation coefficient; S5, setting a spraying timing of a vertical roll lubricating device.

2. The method of controlling the lubrication device of the entry vertical roll of a hot continuous rolling finishing mill according to claim 1, characterized in that, In step S1, the calculation formula of the vertical roll rolling force feedback control model is: In the formula, P1 n P1 represents the current output coefficient of the vertical roll rolling force feedback control model; n-1 PF represents the output coefficient of the vertical roll rolling force feedback control model at the previous moment. n F is the actual rolling force of the vertical roll at the current moment; F is the actual rolling force locked at the moment the vertical roll bites the steel; PF n-1 This represents the actual rolling force of the vertical roll at the previous moment.

3. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 1, characterized in that, In step S2, the calculation formula of the rolling speed feedback control model is: In the formula, P2 n is the output coefficient of the rolling speed feedback control model at the current time; P2 n-1 is the output coefficient of the rolling speed feedback control model at the previous time; P2 n is the actual value of the rolling force of the edger at the current time; S is the actual value of the locked rolling force at the time when the edger bites the steel; P2 n-1 is the actual value of the rolling force of the edger at the previous time.

4. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 1, characterized in that: In step S3, the strip blank center line deviation feedforward control model comprises a vertical roll driving side strip blank center line deviation feedforward control model and a vertical roll working side strip blank center line deviation feedforward control model.

5. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 4, characterized in that: The calculation formula of the vertical roll driving side strip blank center line deviation feedforward control model is: In the formula, P3 n P3 is the current time output coefficient of the strip blank center line deviation feedforward control model on the driven side of the vertical roll; P3 n-1 P3 is the previous time output coefficient of the strip blank center line deviation feedforward control model on the driven side of the vertical roll; P3 n PW is the width deviation value at the acquisition time of the R2 outlet width gauge; W is the set target value of the R2 outlet width of the strip blank; PW n-1 PW is the width deviation value at the previous acquisition time of the R2 outlet width gauge.

6. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 4 or 5, characterized in that: The calculation formula of the vertical roll working side strip blank center line deviation feedforward control model is: In the formula, P4 n P4 is the output coefficient of the vertical roll working side strip blank center line deviation feedforward control model at the current time; P4 n-1 P4 is the output coefficient of the vertical roll working side strip blank center line deviation feedforward control model at the previous time; P4 n PW is the width deviation value of the R2 outlet width gauge at the collection time; W is the set target value of the R2 outlet width of the strip blank; PW n-1 PW is the width deviation value of the R2 outlet width gauge at the previous collection time.

7. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 1, characterized in that, In step S4, the strip temperature characteristic compensation coefficient is specifically as follows: K10=A10i*R10+B10i K11=A11i*R11+B11i K12=A12i*R12+B12i K13=A13i*R13+B13i … K19=A19i*R19+B19i In the above formula list, K1x is a strip edge temperature drop time characteristic layer compensation coefficient; A10-A19 are strip edge temperature drop time characteristic compensation layer proportional gain coefficient 0-9; R10-R19 are strip edge temperature ten-layer interval values; B10-B19 are strip edge temperature drop time characteristic compensation layer adjustment coefficients 0-9; i is the input of temperature difference; K is the output value corresponding to the temperature difference input.

8. The method of controlling the lubrication of the entry vertical roll of a hot continuous rolling finishing mill according to claim 1, characterized in that, In step S5, the control of the spraying timing of the vertical roll lubricating device is specifically as follows: The spraying timing point T-open of the vertical roll lubricating device is set to be opened after the vertical roll steel biting signal is established and after a delay, that is: T-open: F1E LOAD ON + TD When the distance S0 of the tail of the rolled piece from the vertical roll is less than S1+α, the vertical roll lubricating spraying device is closed, that is: T-close: S0 < S1+α Wherein, S1 is the distance of one rotation of the vertical roll, S1=π×R, R is the diameter of the vertical roll; α is a compensation value.

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