JP2020124737A - Mold vibration device, method for producing continuously cast slab and mold vibration method - Google Patents

Abstract

To provide a mold vibration device, a method for producing a continuously cast slab and a mold vibration method capable of realizing the stable continuous casting of a slab by extremely simple means.SOLUTION: A mold vibration device comprises: a vibration generator composed so as to generate vibration in accordance with a prescribed target waveform to a mold composed in such a manner that a poured molten metal is cooled, and a slab is continuously cast; a correction part composed in such a manner that, based on deformation generated in the vibration generator with the vibration of the mold, a plurality of sine wave components composing the target waveform are respectively corrected to generate a plurality of corrected waveforms; and a synthesis part composed in such a manner that the plurality of corrected waveforms are synthesized to generate one synthesized waveform, and the signal of the synthesized waveform is outputted to the vibration generator.SELECTED DRAWING: Figure 3

Description

The present disclosure relates to a mold vibration device, a method for manufacturing a continuously cast slab, and a mold vibration method.

Patent Document 1 discloses a continuous casting facility in which molten metal is poured into a mold to continuously cast slabs. In continuous casting, a technique is known in which a lubricant (for example, mold powder) is added to the molten metal in the mold and the mold is vibrated vertically (casting direction). By vibrating the mold, the lubricant easily flows between the mold and the molten metal, and seizure between the mold and the cast piece can be suppressed. There is also known a technique of increasing the stability of pulling out a slab from a mold by causing the mold to vibrate according to a non-sinusoidal waveform (for example, a vibration in which the descending speed is faster than the ascending speed).

JP, 2003-212156, A

However, the non-sinusoidal wave may be configured to include many harmonic components. If the harmonic component resonates with the vibration mechanism that vibrates the mold, the mold cannot perform a desired operation. Therefore, in Patent Document 1, in order to eliminate the influence of the natural frequency of the vibration mechanism, feedback control is performed based on the state estimation amount of the mold and the actuator. In order to observe the state estimator, it is necessary to construct a state observer and perform complicated matrix operation processing. Controllers capable of such processing are generally expensive.

Therefore, the present disclosure describes a mold vibrating device, a method of manufacturing a continuously cast slab, and a mold oscillating method that can realize stable continuous casting of a slab with an extremely simple method.

Example 1. A mold vibrating device according to one aspect of the present disclosure has a predetermined non-sinusoidal target waveform with respect to a mold configured to cool molten metal poured and continuously cast steel billets. A plurality of sine wave components forming the target waveform are respectively corrected based on the vibration generator configured to generate the corresponding vibration and the deformation generated in the vibration generator due to the vibration of the mold. A correction unit configured to generate a correction waveform, and a combination unit configured to combine a plurality of correction waveforms to generate one combined waveform and output a signal of the combined waveform to the vibration generator. Equipped with. In this case, the correction unit obtains a plurality of correction waveforms in which the deformation caused in the vibration generator is folded in advance. Therefore, when the combined waveform obtained by combining the plurality of correction waveforms is output to the mold via the vibration generator, the combined waveform changes due to the influence of the deformation caused by the vibration generator, and the changed waveform is Acts on the mold. Therefore, ideal vibration can be generated in the mold without using the state estimation amount. As a result, stable continuous casting of the slab can be realized by an extremely simple method.

Example 2. In the apparatus of Example 1, the correction unit includes a plurality of correction waveforms so that each waveform after the respective amplitudes and phases of the correction waveforms are changed substantially matches the corresponding sine wave components. It may be configured to generate the correction waveform. In this case, when the composite waveform obtained by combining the plurality of correction waveforms is output to the mold via the vibration generator, the composite waveform changes to a waveform equivalent to the target waveform due to the influence of deformation by the vibration generator. However, the waveform after this change acts on the mold. Therefore, it becomes possible to vibrate the mold more accurately.

Example 3. In the device of Example 2, the amplitude and the phase may be calculated based on the resonance angular frequency and the damping constant of the vibration generator. In this case, the resonance angular frequency and the damping constant can be obtained relatively easily by experiments or the like. Therefore, it becomes possible to perform the arithmetic processing by the correction unit more easily and at higher speed.

Example 4. In the device of any of Examples 1-3, the plurality of sinusoidal components may include two or three sinusoidal components. In this case, the number of components to be subjected to the correction calculation processing by the correction unit may be small. Therefore, it becomes possible to perform the arithmetic processing by the correction unit more easily and at higher speed.

Example 5. In the device according to any one of Examples 1 to 4, each of the plurality of sine wave components may include an angular frequency that is an integral multiple of the fundamental angular frequency and a different multiple.

Example 6. In the device according to any one of Examples 1 to 5, the vibration generator includes a drive unit capable of linear motion and an arm supported by a rotation fulcrum, and the drive unit and the mold being connected to different positions from each other. May be included.

Example 7. A method for manufacturing a continuously cast slab according to another aspect of the present disclosure, when a vibration corresponding to a predetermined non-sinusoidal target waveform is generated in a mold by a vibration generator, the vibration is generated along with the vibration of the mold. A plurality of sine wave components constituting the target waveform are respectively corrected based on the deformation caused in the machine to generate a plurality of correction waveforms; and a plurality of correction waveforms are combined to generate one combined waveform. And outputting a synthetic waveform signal to a vibration generator to vibrate the mold while pouring molten metal into the mold to continuously cast steel pieces. In this case, the same effect as in Example 1 is obtained.

Example 8. A mold vibrating method according to another aspect of the present disclosure, a mold configured to cool the molten metal poured to continuously cast steel billets, a predetermined non-sinusoidal target waveform. When a corresponding vibration is generated by the vibration generator, a plurality of sinusoidal components forming the target waveform are respectively corrected based on the deformation generated in the vibration generator due to the vibration of the mold, and a plurality of corrected waveforms are generated. That is, the plurality of correction waveforms are combined to generate one combined waveform, and the signal of the combined waveform is output to the vibration generator to vibrate the mold. In this case, the same effect as in Example 1 is obtained.

According to the mold vibrating device, the method for manufacturing a continuously cast slab, and the mold oscillating method according to the present disclosure, stable continuous casting of a slab can be realized by an extremely simple method.

FIG. 1 is a side view schematically showing an example of continuous casting equipment. FIG. 2 is a side view showing an example of the mold vibration device. FIG. 3 is a schematic diagram mainly showing the hardware configuration of the controller. FIG. 4 is a block diagram showing an example of the mold vibration device. FIG. 5 is a diagram showing an example of a target waveform, a plurality of sine wave components, and a plurality of correction waveforms, respectively. FIG. 6 is a diagram showing an example of a composite waveform, a plurality of distortion waveforms, and an example of a vibration waveform. FIG. 7 is a diagram showing an example of models of the vibration generator 110 and the mold 14. FIG. 8A is a diagram showing an example of displacement of the mold when the entire drive arm is a rigid body, and in FIGS. 8B and 8C, the drive arm is partially a rigid body. FIG. 8D is a diagram showing an example of displacement of the mold in the case, and FIG. 8D is a diagram showing an example of displacement of the mold when the drive arm is not a rigid body. FIG. 9A is a graph showing the position command of the piston rod and its actual position according to the embodiment, and FIG. 9B is a graph showing the position command of the mold and its actual position according to the embodiment. Is. FIG. 10A is a graph showing a piston rod position command and its position performance according to a comparative example, and FIG. 10B is a graph showing a mold position command and its position performance according to the comparative example. Is.

Hereinafter, an example of an embodiment according to the present disclosure will be described in more detail with reference to the drawings. In the following description, the same elements or elements having the same function will be denoted by the same reference symbols, without redundant description.

[Construction of continuous casting equipment]
First, the configuration of the continuous casting facility 1 will be described. As shown in FIG. 1, the continuous casting facility 1 includes a ladle 10, a tundish 12, a mold 14, a plurality of cast piece support rolls 16, and a mold vibrating device 100.

The ladle 10 is a container for storing a molten metal (molten metal; molten steel) M. The tundish 12 is arranged below the ladle 10. The tundish 12 is configured to store the molten metal M flowing out from the discharge port of the nozzle provided on the bottom wall of the ladle 10.

The mold 14 is arranged below the tundish 12. The mold 14 is configured to mold the molten metal M poured from a nozzle provided on the bottom wall of the tundish 12 into a predetermined shape while cooling the molten metal M. The plurality of slab support rolls 16 are arranged below the mold 14. The plurality of slab support rolls 16 are configured to convey the slab S drawn from the mold 14 to the downstream side while further cooling.

The slab S immediately after being molded by the mold 14 includes a solidified shell S1 whose surface is solidified and an unsolidified portion S2 filled inside the solidified shell S1. The unsolidified portion S2 is a molten metal in an unsolidified state and has fluidity. The slab S is cooled toward the downstream side, the unsolidified portion S2 is gradually solidified, and the solidified shell S1 grows. That is, as the solidified shell S1 grows, the unsolidified portion S2 shrinks and the thickness of the solidified shell S1 increases. By the time the slab S reaches the slab reduction device (not shown), the slab S is completely solidified.

The mold vibrating device 100 is configured to cause the mold 14 to vibrate according to a predetermined target waveform. As shown in FIG. 2, the mold vibration device 100 includes a vibration generator 110, a sensor 120, a servo valve 130, and a controller 140 (control unit).

The vibration generator 110 includes a drive unit 111 and a link mechanism 112. The drive unit 111 includes a main body 113 and a piston rod 114. The main body 113 is attached to the floor via a rotation fulcrum P1 and can swing around the rotation fulcrum P1. The piston rod 114 is slidably attached to the main body 113. The drive unit 111 may be, for example, a hydraulic cylinder, an electric cylinder in which a ball screw and a servo motor are combined, or a linear motor.

The link mechanism 112 includes a pedestal 115, a drive arm 116 (arm), and an auxiliary arm 117. The frame 115 is fixed to the floor. The drive arm 116 is attached to the lower part of the pedestal 115 via a rotation fulcrum P2 near the center thereof. Therefore, the drive arm 116 can swing around the rotation fulcrum P2. One end of the drive arm 116 is attached to the tip of the piston rod 114 via a rotation fulcrum P3. The other end of the drive arm 116 is connected to the lower part of the mold 14 via a rotation fulcrum P4. One end of the auxiliary arm 117 is attached to the upper part of the pedestal 115 via a rotation fulcrum P5. The other end of the auxiliary arm 117 is attached to the upper portion of the mold 14 via a rotation fulcrum P6.

Since the drive unit 111 and the mold 14 are connected via the link mechanism 112, when the piston rod 114 advances outward, the drive arm 116 swings clockwise around the rotation fulcrum P2, and the mold moves. 14 moves downward. On the other hand, when the piston rod 114 retreats toward the main body 113, the drive arm 116 swings counterclockwise around the rotation fulcrum P2, and the mold 14 moves upward. Therefore, the mold 14 vibrates up and down as the piston rod 114 repeatedly moves forward and backward.

The sensor 120 is configured to detect the position of the piston rod 114 with respect to the body 113. The sensor 120 may be, for example, a direct acting encoder such as a rod sensor. The sensor 120 may be built in the drive unit 111. The data detected by the sensor 120 is transmitted to the controller 140.

The servo valve 130 is configured to open and close the valve based on a control signal from the controller 140 to control the flow direction and flow rate of the hydraulic oil with respect to the main body 113. When the flow direction of the hydraulic oil is changed by the servo valve 130, the moving direction of the piston rod 114 changes. When the flow rate of the hydraulic oil changes due to the servo valve 130, the moving speed of the piston rod 114 changes.

The controller 140 is configured to process the data received from the sensor 120 to control the operation of the servo valve 130.

The hardware of the controller 140 includes, for example, one or a plurality of control computers. As the hardware configuration, the controller 140 has a processor Ctr1 (arithmetic unit), a memory Ctr2 (storage unit), an input port Ctr3 (input unit), and an output port Ctr4 (output as shown in FIG. 3, for example. Part) and. The controller Ctr may be composed of electrical circuits.

The processor Ctr1 cooperates with the memory Ctr2 to execute a program, and inputs/outputs signals through the input port Ctr3 and the output port Ctr4, thereby configuring each functional module described later. That is, the processor Ctr1 is configured to generate an output signal for driving the servo valve 130 based on the input signal from the sensor 120. The memory Ctr2 is configured to store programs, input signals, output signals and the like. The input port Ctr3 is configured to send an input signal from the sensor 120 to the processor Ctr1. The output port Ctr4 is configured to send the output signal generated by the processor Ctr1 to the servo valve 130.

As shown in FIG. 4, the controller 140 includes, as functional modules, a target waveform setting unit 141, a fundamental wave generator 142, a corrector 143 (correction unit), a combiner 144 (composition unit), and a comparator. 145 and an amplifier 146. It should be noted that these functional modules may be realized by software by a program, or may be realized by an electric circuit element (for example, a logic circuit) or an integrated circuit in which the electric circuit element is integrated.

The target waveform setting device 141 is configured to set a predetermined target waveform W1 condition for vibrating the mold 14 based on, for example, an input from an operator. The condition of the target waveform W1 is output to the fundamental wave generator 142. The target waveform W1 may be non-sinusoidal. As shown in FIG. 5A, the target waveform W1 may be, for example, a waveform that causes the mold 14 to vibrate such that the descending speed of the mold 14 is faster than the ascending speed. When the target waveform W1 is a non-sinusoidal wave, the conditions of the target waveform W1 may include, for example, the amplitude and the non-sine rate or the distortion coefficient. In the present specification, the term “non-sine rate” refers to the ratio of the variation time from the valley to the peak of the non-sinusoidal wave in the half cycle to the variation time from the peak to the valley of the non-sinusoidal wave in the half cycle. To do. In the present specification, the distortion coefficient means the ratio of the amplitude of cos2ωt to the amplitude of sinωt.

The fundamental wave generator 142 has a plurality of sine wave components W2 ₁ , W2 ₂ ,..., W2 _N (where N is an integer of 2 or more) based on the condition of the target waveform W1 set by the target waveform setting unit 141. ) Is configured to generate. The generated plurality of sine wave components W2 ₁ , W2 ₂ ,..., W2 _N are output to the corrector 143. Of these, an arbitrary sine wave component W2 _n (where n is a natural number of 1 to N) becomes equal to, for example, the n-th term (n-th harmonic) when the target waveform W1 is expanded by Fourier series. The number of the plurality of sine wave components forming the target waveform W1 is not particularly limited, but may be two or three, for example. FIG. 5B shows an example in which the target waveform W1 is decomposed into _three sine wave components W2 ₁ , W2 ₂ , W2 ₃ .

The corrector 143 corrects each of the plurality of sine wave components W2 ₁ , W2 ₂ ,..., W2 _N based on the deformation (for example, the deflection of the drive arm 116) generated in the link mechanism 112 due to the vibration of the mold 14. By doing so, a plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N are configured to be generated. The plurality of generated correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N are output to the combiner 144.

By the way, when the composite waveform W4 of the plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N is input to the link mechanism 112, the link mechanism 112 is deformed due to the vibration of the mold 14. Therefore, a plurality of correction waveform _{W3 1,} W3 2 by the deformation, · · ·, _{W3 N} plurality of distorted waveform distorted respectively _{W5 1, W5 2, ···,} W5 N is molded from the vibration generator 110 14 Is output to. Accordingly, the corrector 143, a plurality of distorted waveform _{W5 1, W5 2, ···,} W5 N each multiple sinusoidal components _{W2 1, W2 2, ···,} W2 N substantially more to conform The correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N may be generated. FIG. 5C shows an example of _three correction waveforms W3 ₁ , W3 ₂ , W3 ₃ in which the three sine wave components W2 ₁ , W2 ₂ , W2 ₃ are respectively corrected in consideration of the deformation of the vibration generator 110. Show. A more detailed correction calculation method will be described later.

The combiner 144 is configured to combine the plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N generated by the corrector 143 to generate one combined waveform W4. The generated composite waveform W4 is output to the comparator 145. The composite waveform W4 may be, for example, the sum of a plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N (W4=W3 ₁ +W3 ₂ +... +W3 _N ). FIG. 6A shows an example of a combined waveform W4 in which the _three correction waveforms W3 ₁ , W3 ₂ , and W3 ₃ are combined.

The comparator 145 is configured to calculate a deviation E between the current value (target value) X of the composite waveform W4 and the current value Y of the position of the piston rod 114 detected by the sensor 120. That is, the comparator 145 calculates the deviation E by subtracting the current value Y from the target value X (E=XY). The calculated deviation E is output to the amplifier 146.

The amplifier 146 is configured to calculate the output value Q by multiplying the deviation E calculated by the comparator 145 by a predetermined proportional gain Kp (Q=Kp×E). The output value Q calculated by the amplifier 146 is output to the servo valve 130. The piston rod 114 of the drive unit 111 advances and retreats (elevates) by the servo valve 130 that operates based on the output value Q, and the mold 14 vibrates via the link mechanism 112. At this time, as described above, a plurality of correction waveform _W3 1 constituting the composite waveform _W4, W3 2, · · ·, distorted _{W3 N,} a plurality of distorted waveform _{W5 1, W5 2, ···,} W5 N Acts on the mold 14.

FIG. 6B shows an example of _three distortion waveforms W5 ₁ , W5 ₂ , W5 ₃ after the three correction waveforms W3 ₁ , W3 ₂ , W3 ₃ are distorted through the link mechanism 112. The three distortion waveforms W5 ₁ , W5 ₂ , and W5 ₃ substantially match the _three sine wave components W2 ₁ , W2 ₂ , and W2 ₃ , respectively. Further, in FIG. 6 (c), shows an example of a vibration waveform W6 of the mold 14 when _{the W5 1,} W5 2, W5 ₃ of the composite wave three distorted waveform is applied to the mold 14. The vibration waveform W6 substantially matches the target waveform W1.

[Calculation method of correction]
(1) Modeling of Vibration Generator 110 and Mold 14 A calculation method for obtaining a correction waveform W3 _{n in} which the distortion waveform W5 _n substantially matches the sine wave component W2 _n will be described. For explanation, the vibration generator 110 and the mold 14 are modeled as shown in FIG. 7. Specifically, the parameters L ₁ , L ₂ , A ₁ , A ₂ , and m ₂ are defined as follows, respectively. The portion of the drive arm 116 between one end on the drive unit 111 side (input side) and the rotation fulcrum P2 is referred to as “drive arm 116 ₁ ”, and the other end on the mold 14 side (output side) and the rotation fulcrum P2. The portion of the drive arm 116 between the _two is referred to as a “drive arm 116 ₂ ”. Further, the upward direction of FIG. 7 is positive.

L ₁ : Length of drive arm 116 ₁ [m]
L ₂ : Length of drive arm 116 ₂ [m]
A ₁ : Amplitude [m] of piston rod 114 (one end of drive arm 116)
A ₂ : Amplitude [m] of the mold 14 (the other end of the drive arm 116)
m ₂ : mass of the mold 14

また、モールド１４の変位をｘ_２とするとｘ_１：Ｌ_１＝ｘ_２：Ｌ_２が成り立つので、変位ｘ_２について式２が得られる。なお、式２において、Ａ_２は変位ｘ_２の振幅である。

ここで、式２より、Ａ_２について式３が成り立つ。

(2) When the entire drive arm 116 is a rigid body First, it is assumed that the entire drive arm 116 is a rigid body and does not deform (see FIG. 8A). In this case, when a sinusoidal vibration is applied to one end of the drive arm 116 by the drive unit 111, the displacement x ₁ of the one end is expressed by Expression 1. In Equation 1, the parameters ω, t, θ ₁ are
ω: Angular frequency [rad/sec]
t: time [sec]
θ ₁ : initial phase of displacement x ₁ [rad]
Is.

Further, assuming that the displacement of the mold 14 is x ₂ , x ₁ :L ₁ =x ₂ :L ₂ is established, and therefore the equation 2 is obtained for the displacement x ₂ . Note that in Expression 2, A ₂ is the amplitude of the displacement x ₂ .

Here, from Expression 2, Expression 3 holds for A ₂ .

ここで、ａ_２は、式２を２階微分することにより得られる（式５参照）。

式５を式４に代入すると、式６が得られる。

(3) When the drive arm 116 ₁ is not a rigid body Subsequently, it is assumed that the drive arm 116 ₂ is a rigid body, but the drive arm 116 ₁ is not a rigid body but has flexibility (a leaf spring) (FIG. 8). (See (b)). In this case, the equation of motion of the drive arm 116 is expressed by Equation 4. In Equation 4, the parameters k ₁ , c ₁ , a ₂ , a ₂₁ , v ₂₁ , and x ₂₁ are
k ₁ : Spring constant of drive arm 116 ₁ [N/m]
c ₁ : damping coefficient of drive arm 116 ₁ [N·sec/m]
a ₂ : acceleration generated in the mold 14 by the operation of the driving unit 111 [m/sec ² ]
a ₂₁ : Acceleration [m/sec ² ] generated in the mold 14 by the deflection of the driving arm 116 ₁ .
v ₂₁ : Velocity [m/sec] generated in the mold 14 by the deflection of the drive arm 116 ₁ .
x ₂₁ : Displacement generated in the mold 14 due to the deflection of the drive arm 116 ₁ [m]
Is.

Here, a ₂ is obtained by performing the second-order differentiation of Expression 2 (see Expression 5).

Substituting equation 5 into equation 4 yields equation 6.

ただし、減衰定数ｈ_１は測定によって得られる値であり、共振角周波数ω_ｎ１及びパラメータＺ_１はそれぞれ式１０及び式１１にて定義される。

Equation 6 is a form of the equation of motion for forced vibration, and when sufficient time elapses (t→∞), the general solution of the equation of motion for forced vibration becomes equal to the special solution. Therefore, parameters A ₂₁ , φ ₁ , h ₁ , and ω _n1 are respectively set as A ₂₁ : amplitude [m] of displacement x ₂₁ .
φ ₁ : delay phase of displacement x ₂₁ [rad]
h ₁ : damping constant (damping ratio)
ω _n1 : Resonance angular frequency [rad/sec]
Then, the special solution of Expression 6 is expressed by Expression 7, the amplitude A ₂₁ is expressed by Expression 8, and the delay phase φ ₁ is expressed by Expression 9. However, it is defined that θ ₂₁ =θ ₁ −φ ₁ .

However, the damping constant h ₁ is a value obtained by measurement, and the resonance angular frequency ω _n1 and the parameter Z ₁ are defined by Equation 10 and Equation 11, respectively.

式５を式１２に代入すると、式１３が得られる。

(4) When the drive arm 116 ₂ is not a rigid body Subsequently, it is assumed that the drive arm 116 ₁ is a rigid body, but the drive arm 116 ₂ is not a rigid body but has flexibility (a leaf spring) (FIG. 8). (See (c)). In this case, the equation of motion of the drive arm 116 is expressed by Equation 12. In Expression 12, the parameters k ₂ , c ₂ , a ₂ , a ₂₂ , v ₂₂ , and x ₂₂ are
k ₂ : Spring constant [N/m] of the drive arm 116 ₂ .
c ₂ : damping coefficient of drive arm 116 ₂ [N·sec/m]
a ₂ : acceleration generated in the mold 14 by the operation of the driving unit 111 [m/sec ² ]
a _22: acceleration generated in the mold 14 by the deflection of the drive arm 116 ₂ [m / ^sec _2]
v ₂₂ : Velocity [m/sec] generated in the mold 14 by the deflection of the drive arm 116 ₂ .
x ₂₂ : Displacement [m] generated in the mold 14 due to the deflection of the drive arm 116 ₂ .
Is.

Substituting equation 5 into equation 12 yields equation 13.

ただし、減衰定数ｈ_２は測定によって得られる値であり、共振角周波数ω_ｎ２及びパラメータＺ_２はそれぞれ式１７及び式１８にて定義される。

Equation 13 is a form of the equation of motion for forced vibration, and when sufficient time has passed (t→∞), the general solution of the equation of motion for forced vibration becomes equal to the special solution. Therefore, the parameters A ₂₂ , φ ₂ , h ₂ , and ω _n2 are respectively set to A ₂₂ : the amplitude [m] of the displacement x ₂₂ .
φ ₂ : delay phase of displacement x ₂₂ [rad]
h _2: decay constant (damping ratio)
ω _n2 : Resonance angular frequency [rad/sec]
Then, the special solution of Expression 13 is expressed by Expression 14, the amplitude A ₂₂ is expressed by Expression 15, and the delay phase φ ₂ is expressed by Expression 16. However, it is defined that θ ₂₂ =θ ₁ −φ ₂ .

However, the damping constant h ₂ is a value obtained by measurement, and the resonance angular frequency ω _n2 and the parameter Z ₂ are defined by Equations 17 and 18, respectively.

(5) When both drive arms 116 ₁ and 116 ₂ are not rigid bodies Next, the case where both drive arms 116 ₁ and 116 ₂ are not rigid bodies (actual drive arms 116) will be examined (see FIG. 8D). .. That is, it is assumed that the entire drive arm 116 is a leaf spring. The actual displacement x ₀ of the mold 14 at this time is, as shown in Expression 19, the displacement x ₂ of the mold 14 when the drive arm 116 does not bend and the displacement 14 of the mold 14 when the drive arm 116 ₁ bends. It can be expressed by superposing the displacement x ₂₁ and the displacement x ₂₂ of the mold 14 when the drive arm 116 ₂ bends.

式１９に、式２、式７、式１４及び式２０を代入すると、式２１が得られる。

ここで、加法定理を用いて式２１の右辺を展開すると、式２２が得られる。

ただし、式２２において、パラメータａ，ｂは以下の式２３及び式２４とおりである。

When the amplitude of the displacement _{x 0} and _{A 0,} the initial phase of the displacement _{x 0} and theta _0, the displacement _{x 0} is expressed by Equation 20.

By substituting the equation 2, the equation 7, the equation 14, and the equation 20 into the equation 19, the equation 21 is obtained.

Here, when the right side of Expression 21 is expanded using the addition theorem, Expression 22 is obtained.

However, in Expression 22, the parameters a and b are as shown in Expression 23 and Expression 24 below.

From Equations 23 and 24, A ₀ , sin θ ₀ , and cos θ ₀ are represented by Equations 25 to 27, respectively.

式２８をＡ_１について解くと、式２９が得られる。

By substituting Equation 3, Equation 8, Equation 9, Equation 15, and Equation 16 into Equation 25, Equation 28 is obtained.

Solving Equation 28 for A ₁ , yields Equation 29.

式３０に式３、式８、式９、式１５及び式１６を代入して、θ_１について整理すると、式３１が得られる。

On the other hand, when the initial phase θ ₀ on the mold 14 side is used as the reference (θ ₀ =0), sin θ ₀ =0, and therefore Equation 30 is obtained from Equation 27.

By substituting the equation 3, the equation 8, the equation 9, the equation 15, and the equation 16 into the equation 30 and rearranging about θ ₁ , the equation 31 is obtained.

By the way, in Expression 9, Expression 11, Expression 16, and Expression 18, ω is the frequency of the ideal vibration to be generated in the mold 14, and is determined by the setting. In Expression 29, A ₀ is the ideal amplitude of vibration to be generated in the mold 14, and is determined by the setting. Therefore, according to Expression 9, Expression 11, Expression 16, Expression 18, Expression 29, and Expression 31, the unknowns included in A ₁ and θ ₁ are four, ω _n1 , ω _n2 , h ₁ , and h _2. .. From the above, by substituting these variables ω _n1 , ω _n2 , h ₁ , and h ₂ and the ideal vibration amplitude A ₀ and the frequency ω to be generated in the mold 14 into Equation 29 and Equation 31, respectively. It is possible to obtain the amplitude A ₁ and the initial phase θ ₁ of the piston rod 114 for generating ideal vibration in the mold 14.

(6) Method of Measuring Unknowns ω _n1 , ω _n2 , h ₁ , h ₂ The unknowns ω _n1 , ω _n2 may be measured as follows, for example. First, the actual vibration generator 110 and the mold 14 are prepared. Next, in a state where the vibration generator 110 and the mold 14 are stationary, the controller 140 outputs an impulse signal to the servo valve 130. That is, the piston rod 114 is operated with a predetermined stroke amount in a very short time. Next, the displacement generated in the mold 14 by the input of the impulse signal is measured. Next, the cycle of damping free vibration of the mold 14 is measured from the measured value of the displacement of the mold 14. As a result, the unknowns ω _n1 and ω _n2 are obtained. On the other hand, the unknowns h ₁ and h ₂ may be measured by various known methods such as the half-width method, the attenuation rate method, and the impedance method.

[Method of manufacturing slab]
Subsequently, a method of manufacturing the cast slab S (a method of vibrating the mold 14) will be described. First, the resonance angular frequency ω _n and the damping constant h of the vibration generator 110 and the actual mold 14 are measured in advance. Next, the operator inputs the ideal vibration data to be generated in the mold 14 into the controller 140. The target waveform setting unit 141 generates a target waveform W1 corresponding to the vibration based on the input, and outputs the target waveform W1 to the fundamental wave generator 142.

Next, the fundamental wave generator 142 decomposes the target waveform W1 into a plurality of sinusoidal wave components W2 ₁ , W2 ₂ ,..., W2 _N. As a result, the amplitude ⁿ A ₀ and the angular frequency ⁿ ω of the arbitrary sine wave component W2 _n can be obtained. The fundamental wave generator 142 outputs these sine wave components W2 ₁ , W2 ₂ ,..., W2 _N to the corrector 143. When the basic angular frequency is ω ₀ , the angular frequency ⁿ ω may be an integral multiple of the basic angular frequency ω ₀ as shown in Expression 32. The magnitude of the basic angular frequency ω ₀ may be set according to the vibration generator 110 and the actual machine of the mold 14. For example, the value of ω ₀ /2π may be set to 6 Hz or less.

Next, the corrector 143 uses the amplitudes and angular frequencies of the sine wave components W2 ₁ , W2 ₂ ,..., W2 _N , the resonance angular frequency ω _n, and the damping constant h to calculate the sine wave component W2 ₁ , Each of W2 ₂ ,..., W2 _N is corrected to generate a plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N. For example, the amplitude ⁿ A ₁ and the initial phase ⁿ θ ₁ of the arbitrary correction waveform W3 _n can be calculated from Expression 29 and Expression 31, respectively. Therefore, the displacement ⁿ x ₁ of the arbitrary correction waveform W3 _n is obtained by Expression 33 by substituting the amplitude ⁿ A ₁ and the initial phase ⁿ θ ₁ in Expression ₁ . The corrector 143 outputs the generated plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N to the combiner 144.

Next, the combiner 144 combines the plurality of corrected waveforms W3 ₁ , W3 ₂ ,..., W3 _N to generate one combined waveform W4. The displacement x ₁ of the composite waveform W4 is expressed by Expression 34 using Expression 33.

The combined waveform W4 generated by the combiner 144 is output to the servo valve 130 through the comparison processing by the comparator 145 and the amplification processing by the amplifier 146. As a result, the servo valve 130 operates the drive unit 111, and vibration according to the composite waveform W4 acts on the link mechanism 112. In the process in which the vibration propagates through the link mechanism 112, the synthetic waveform W4 is distorted due to the influence of the deformation of the drive arm 116, and the vibration corresponding to the target waveform W1 is generated in the mold 14. When the molten metal M is poured from the tundish 12 to the vibrating mold 14 according to the target waveform W1, the molten metal M is cooled by the mold 14 and shaped into a predetermined shape. When the slab S drawn from the mold 14 is further cooled and completely solidified, the slab S is completed.

[Action]
According to the above example, the corrector 143 obtains a plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _{N in} which the deformation caused in the vibration generator 110 is folded in advance. Therefore, when the composite waveform W4, which is a composite of the plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N, is output to the mold 14 via the vibration generator 110, the deformation by the vibration generator 110 is suppressed. synthesized waveform W4 is changed under the influence, this change after the waveform (s distorted waveform _{W5 1,} W5 2, · · ·, the composite wave of _{W5 N)} is applied to the mold 14. Therefore, ideal vibration can be generated in the mold 14 without using the state estimation amount. As a result, stable continuous casting of the slab S can be realized by an extremely simple method. Further, in this case, since the arithmetic processing by the corrector 143 is extremely simplified, the arithmetic processing can be performed at low cost and high speed by using, for example, a commercially available controller or the like.

According to the above example, even if the angular frequencies ω of the plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N are values close to the natural frequency, the influence of resonance is suppressed by the correction, It is possible to suppress abnormal vibration of the mold 14.

According to the above example, the amplitude A ₁ and the phase θ ₁ are calculated based on the resonance angular frequencies ω _n1 and ω _n2 of the vibration generator 110 (driving arm 116) and the damping constants h ₁ and h ₂ . The resonance angular frequencies ω _n1 and ω _n2 and the damping constants h ₁ and h ₂ can be obtained relatively easily by experiments or the like. Therefore, the arithmetic processing by the corrector 143 can be performed more easily and at high speed.

In the above example, the actual displacement x ₀ of the mold 14 in the case where the drive arms 116 ₁ and 116 ₂ are not rigid bodies has been examined, but in consideration of the case where bending or distortion occurs in other elements, The displacement X ₀ may be calculated. Examples of the other elements include the piston rod 114, the pedestal 115, the auxiliary arm 117, and the like.

[Test results]
In order to confirm that the mold 14 vibrates according to the target waveform W1 when the mold 14 is vibrated using the vibration generator 110 and the mold 14 according to the above example, a test was performed using an actual machine. (Example). Specifically, the position command waveform of the mold 14 (see the broken line in FIG. 9B) was used as the target waveform W1 and after correction by the corrector 143, the mold 14 was vibrated and the position change of the mold 14 was measured. As a result, the waveform of the actual position of the mold 14 (see the solid line in FIG. 9B) vibrated according to the waveform corresponding to the composite waveform W4 (see the broken line in FIG. 9A) is the position command of the mold 14. It almost coincided with the waveform (see the broken line in FIG. 9B). Therefore, it was confirmed that the mold 14 vibrates according to the target waveform W1.

In this test, the resonance frequency ω _n /2π was 18.5 Hz and the damping constant h was 0.087. In the fundamental wave generator 142, the fundamental frequency is set to 370 cpm (cycles per minute), and three fundamental frequencies 370 cpm (≈6.17 Hz), 740 cpm (≈12.33 Hz), and 1110 cpm (=18.5 Hz) are provided. A sine wave component was generated from the target waveform W1.

On the other hand, the same test was performed without correction by the corrector 143 (comparative example). Specifically, the mold 14 is operated by operating the piston rod 114 with a waveform (see the broken line in FIG. 10A) obtained by reversing the waveform of the position command of the mold 14 (see the broken line in FIG. 10B). The position change of was measured. As a result, the waveform of the actual position of the mold 14 (see the solid line in FIG. 10B) was greatly distorted and did not match the waveform of the position command of the mold 14 (see the broken line in FIG. 10B).

[Modification]
Although the embodiments according to the present disclosure have been described in detail above, various modifications may be made to the above embodiments without departing from the scope of the claims and the gist thereof. For example, the compensator 143 includes a plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N after the amplitude and phase of each of the plurality of correction waveforms W3 ₁ , W3 ₂ ,... the distorted waveform _{W5 1, W5 2, ···,} W5 N is a plurality of sinusoidal components _{W2 1,} W2 2, · · ·, to substantially coincide with _{W2 N,} a plurality of correction waveform _W3 1, W3 ₂ ,..., W3 _N may be configured to be generated. In this case, when the composite waveform W4, which is a composite of the plurality of correction waveforms W3 ₁ , W3 ₂ ,..., W3 _N, is output to the mold 14 via the vibration generator 110, the influence of deformation by the vibration generator 110 is exerted. In response, the composite waveform W4 changes to a waveform W6 equivalent to the target waveform W1, and the changed waveform W6 acts on the mold 14. Therefore, the mold 14 can be vibrated more accurately.

The fundamental wave generator 142 may be configured to decompose the target waveform W1 into 2 to 5 sine wave components, and is configured to decompose the target waveform W1 into 2 to 3 sine wave components. May be. In this case, the number of components to be the target of the correction calculation processing by the corrector 143 is small. Therefore, the arithmetic processing by the corrector 143 can be performed more easily and at high speed.

The angular frequency of each of the plurality of sine wave components W2 ₁ , W2 ₂ ,..., W2 _N may be a value equal to or less than twice the resonance angular frequency.

In the above example, the drive unit 111 is connected to the mold 14 via the link mechanism 112, and the drive arm 116 swings as the piston rod 114 moves back and forth, causing the mold 14 to vibrate up and down. .. However, the drive unit 111 may be directly or indirectly connected to the mold 14 not via the link mechanism 112. In this case, advance/retreat of the piston rod 114 immediately raises/lowers the mold 14.

DESCRIPTION OF SYMBOLS 1... Continuous casting equipment, 14... Mold, 100... Mold vibration device, 110... Vibration generator, 111... Drive part, 112... Link mechanism, 116... Drive arm (arm), 120... Sensor, 130... Servo valve, 140 ... Controller (control unit), 141... Target waveform setting device, 142... Fundamental wave generator, 143... Corrector (correction unit), 144... Synthesizer (Synthesis unit), M... Molten metal, P2... Rotation fulcrum, S ... slab, W1 ... target waveform, W2 ₁ , W2 ₂ , ..., W2 _N ... sine wave component, W3 ₁ , W3 ₂ , ..., W3 _N ... correction waveform, W4 ... composite waveform.

Claims (8)

Vibration generation configured to generate vibration according to a predetermined non-sinusoidal target waveform for a mold configured to cool molten metal poured and continuously cast steel slabs Machine,
A correction unit configured to generate a plurality of correction waveforms by correcting a plurality of sine wave components forming the target waveform based on the deformation of the vibration generator caused by the vibration of the mold. ,
A mold vibrating device, which is configured to combine the plurality of corrected waveforms to generate one combined waveform and output a signal of the combined waveform to the vibration generator. The correction unit corrects the plurality of correction waveforms such that the waveforms after the respective amplitudes and phases of the correction waveforms are changed due to the deformation are substantially equal to the corresponding sinusoidal wave components. The apparatus according to claim 1, wherein the apparatus is configured to generate a waveform. The apparatus according to claim 2, wherein the amplitude and the phase are calculated based on a resonance angular frequency and a damping constant of the vibration generator. 4. The apparatus according to any one of claims 1-3, wherein the plurality of sinusoidal components comprises two or three sinusoidal components. The apparatus according to claim 1, wherein each of the plurality of sine wave components includes an angular frequency that is an integer multiple of a fundamental angular frequency and is a multiple different from each other. The vibration generator,
A drive unit capable of linear motion,
The apparatus according to any one of claims 1 to 5, further comprising: an arm supported by a rotation fulcrum, and an arm connected to different portions of the drive unit and the mold. A plurality of the target waveforms are formed based on the deformation of the vibration generator caused by the vibration of the mold when the vibration is generated by the vibration generator according to the predetermined non-sinusoidal target waveform with respect to the mold. Compensate each sine wave component of to generate multiple compensation waveforms,
Combining the plurality of correction waveforms to generate one combined waveform,
Outputting the signal of the composite waveform to the vibration generator and vibrating the mold, pouring molten metal into the mold, and continuously casting a steel slab, of a continuous cast slab Production method. For the mold configured to cool the molten metal poured and continuously cast the steel slab, when causing a vibration according to a predetermined non-sinusoidal target waveform by the vibration generator, Based on the deformation caused in the vibration generator with the vibration of the mold, respectively correcting a plurality of sine wave components constituting the target waveform, to generate a plurality of correction waveforms,
Combining the plurality of correction waveforms to generate one combined waveform,
Outputting a signal of the composite waveform to the vibration generator to vibrate the mold.

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