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Abstract

The driving mode by servo motor is a new driving mode and can realize oscillations of continuous casting mold as expected. The oscillation system of continuous casting mold driven by servo motor is a complicated nonlinear system. Therefore, it is necessary to establish an accurate model for the system before performing simulation experiments. Based on the oscillation platform system of continuous casting mold driven by servo motor in the laboratory, the nonlinear models of servo motor speed system and mechanical transmission parts were respectively established and the viscous friction coefficient, moment of inertia, and load torque were identified. Then, the nonlinear mathematical model of the whole oscillation platform of continuous casting mold driven by servo motor was obtained. The comparison results between the simulated output curves of established model and the measured output curves of actual system under the same input signals indicated that the simulated curves were almost in accord with the measured curves. Therefore, the proposed model can reflect the capabilities of practical system and lay a good foundation for subsequent research on simulation experiments and accurate control of continuous casting mold oscillation driven by servo motor.

Keywords

Continuous casting mold driving mode by servo motor nonlinear system modeling parameter identification

Introduction

Continuous casting is an important step in steel production. Continuous casting mold is forming equipment and its properties are vital to slab quality and production of continuous casting machine.^1,2

The evolution of the continuous casting mold oscillation technology³ is mainly divided into two stages: the development stage of oscillation waveform and the development stage of oscillation device.⁴ In the early stage, rectangular oscillation was adopted, and then gradually developed into trapezoidal oscillation, sinusoidal oscillation, and non-sinusoidal oscillation.^5–7 Li et al.⁸ introduced the non-sinusoidal oscillation regulation and analyzed the advantage of non-sinusoidal oscillation compared to sinusoidal oscillation. The most remarkable feature of non-sinusoidal oscillation is that the rising time is much longer than lowering time, thus increasing casting speed and improving slab quality. The non-sinusoidal oscillation of continuous casting mold is one of the key technologies for efficient continuous casting.^9,10

Three driving modes, including mechanical drive, hydraulic drive, and servo electric-cylinder drive, are mainly adopted to drive non-sinusoidal oscillation of continuous casting mold in industrial fields. Non-sinusoidal oscillation devices driven by machine, such as non-circular gear (ellipse gear) device¹¹ and anti-parallel four-bar linkage device,^12,13 have been developed in China. Mold non-sinusoidal oscillation device driven by electro-hydraulic servo system has been widely applied at home and abroad.^14,15

In this article, servo motor was adopted to realize non-sinusoidal/sinusoidal oscillation of continuous casting mold.^16,17 Servo motor replaced normal AC motor¹⁸ to run continuously at single-direction variable angular velocity and realize oscillation through reducer and eccentric shaft linkage mechanism. Based on the above principles, we set up the oscillation platform system of continuous casting mold driven by servo motor in the laboratory. After previous studies on non-sinusoidal waveform, parameter selection principle,^19,20 and relevant modeling methods,^21,22 we established a mathematical model to describe the properties of the system better for further simulation experiments of continuous casting mold.

In this article, we first introduced the overall structure of the system comprising servo motor speed system and mechanical transmission parts, and then established the nonlinear models of these two parts, and identified the viscous friction coefficient, moment of inertia, and load torque with the least square method to obtain the relatively accurate mechanism model of the whole continuous casting mold oscillation system. Finally, we compared the simulated output curves of the established model with the actual system output curves under the same input signals, analyzed the errors, and validated the effectiveness of the model.

Structure of oscillation platform of continuous casting mold driven by servo motor

Figure 1 shows the speed control cabinet and oscillation platform of continuous casting mold in the laboratory. The servo motor is permanent magnet synchronous motor (Siemens 1FT6134-6SB71). The servo motor and corresponding servo drive unit S120 forms the speed loop and current loop.

Figure 1.

Speed control cabinet and oscillation platform of continuous casting mold.

Figure 2 shows the structure diagram of oscillation platform of continuous casting mold driven by servo motor composed of servo motor, coupling, reducer, eccentric shaft, linkage, mold, shaker arm, and so on. The control structure is shown in Figure 3. The servo motor speed system is connected with mechanical transmission parts through the reducer.

Figure 2.

Structure diagram of oscillation platform of continuous casting mold driven by servo motor.

Figure 3.

Control structure of the oscillation platform of continuous casting mold driven by servo motor.

In Figure 3, the given speed of servo motor $n_{r}$ is the system input and the displacement of mold $S_{m}$ is the system output. The rest physical parameters are defined as follows: n is the actual motor speed, $ω$ is the motor angular velocity, $θ_{m}$ is the motor output angular displacement, $θ$ is the angular displacement of eccentric shaft, and $S_{e}$ is the vertical displacement of eccentric shaft. The mathematical models of servo motor speed system and mechanism transmission parts are respectively established below.

Modeling for oscillation platform of continuous casting mold driven by servo motor

Modeling of the servo motor system and parameter identification

According to the vector control principle, in the $d - q$ coordinate axis (direct axis and quadrature axis), the mathematical model of permanent magnet synchronous motor is expressed as

{\begin{matrix} {\overset{\cdot}{i}}_{d} = - \frac{R}{L} i_{d} + \frac{U_{d}}{L} + p ω i_{q} \\ {\overset{\cdot}{i}}_{q} = - \frac{R}{L} i_{q} + \frac{U_{q}}{L} - p ω i_{d} - \frac{p φ_{f} ω}{L} \\ \overset{\cdot}{ω} = - \frac{B}{J} ω + \frac{3 p φ_{f}}{2 J} i_{q} - \frac{T_{L}}{J} \\ n = \frac{60}{2 π} ω \end{matrix}

(1)

where $U_{d}$ and $U_{q}$ are, respectively, the stator voltage components on $d - q$ axis $(V)$ ; $i_{d}$ and $i_{q}$ are, respectively, the stator current components on $d - q$ axis $(A)$ ; L is the stator inductance on $d - q$ axis $(mH)$ ; R is the stator resistance $(Ω)$ ; $φ_{f}$ is the permanent magnet flux linkage $(Wb)$ ; p is the number of pole pairs; J is the total moment of inertia (including motor itself and oscillation platform weight on motor shaft) $(kg m^{2})$ ; B is the viscous friction coefficient $(Nms / rad)$ ; and $T_{L}$ is the load torque $(N m)$ .

The parameters of servo motor are provided in Table 1. Moment of inertia of the motor itself is $0.0625 kg m^{2}$ . According to $K_{T} = 3 p φ_{f} / 2$ , $φ_{f}$ is calculated to be $0.704 Wb$ .

Table 1.

Parameters of servo motor.

Technical parameters	Values	Units
Rated power ( $P_{N}$ )	20.4	$kW$
Rated current ( $I_{N}$ )	45	$A$
Rated speed ( $n_{N}$ )	1500	$r / \min$
Stator inductance (L)	4.6	$mH$
Stator resistance (R)	0.14	$Ω$
Torque current ratio ( $K_{T}$ )	3.17	$Nm / A$
Pole pairs (p)	3	–

The identification process of parameters J and B is described below.

The motor torque equation is expressed as

K_{T} i_{q} - T_{L} - B ω = J \frac{d ω}{dt}

(2)

The characteristic curves of load torque are basically the same at two similar rotation speeds. Therefore, equation (2) can be transformed into equation (3)

K_{T} Δ i_{q} (k) = J Δ \overset{\cdot}{ω} (k) + B Δ ω (k) + e (k)

(3)

where $Δ i_{q} (k) = i_{1 q} (k) - i_{2 q} (k)$ , $Δ \overset{\cdot}{ω} (k) = {\overset{\cdot}{ω}}_{1} (k) - {\overset{\cdot}{ω}}_{2} (k)$ , and $Δ ω (k) = ω_{1} (k) - ω_{2} (k)$ are, respectively, the current difference, angular acceleration difference, and angular speed difference between two similar speeds. $Z = [K_{T} Δ i_{q} (1), K_{T} Δ i_{q} (2), \dots, K_{T} Δ i_{q} (N)]^{T}$ is defined as the input measured value vector, $ϕ = [\begin{matrix} Δ \overset{\cdot}{ω} (1), Δ \overset{\cdot}{ω} (2), \dots, Δ \overset{\cdot}{ω} (N) \\ Δ ω (1), Δ ω (2), \dots, Δ ω (N) \end{matrix}]$ is defined as the parameter measured value vector, $X = (J, B)^{T}$ is defined as the parameter vector to be identified.

Then, equation (3) is converted into equation (4)

Z = ϕ^{T} X + e

(4)

According to the least square estimation theory, the estimated values of J and B are

\hat{X} = (ϕ^{T} ϕ)^{- 1} ϕ^{T} Z

(5)

The expected speed of motor is expressed as

n_{r} (t) = \frac{60 i ω_{0}}{2 π} (1 - A \cos (ω_{0} t))

(6)

where reduction ratio $i = 5.1145$ ; $ω_{0} = 2 π f / 60$ ; f is the oscillation frequency of mold and varies from 0 to 200 (times/min); $A = \frac{π α}{2 \sin (\frac{π}{2} (1 + α))}$ , where $α$ is the waveform deviation rate and varies from −0.4 to +0.4. When $α = 0 (A = 0)$ , the mold is sinusoidal oscillation; when $α \neq 0 (A \neq 0)$ , the mold is non-sinusoidal oscillation.

The parameter identification process is introduced as follows:

Viscous friction coefficient can be calculated with the motor current (current of q axis) data (shown in Figure 4) of two similar constant speed (sinusoidal oscillation, $f = 90 times / min$ and $f = 120 times / \min$ ). The sampling period is 0.004 s; total sampling time is 3 s; and the sampling number is $N = 750$ . According to equation (5), the identification value of viscous friction coefficient is $B = 0.0706 Nms / rad$ .

On the basis of viscous friction coefficient value, total moment of inertia is calculated by motor current data (shown as Figure 5) of two similar time-varying speed (non-sinusoidal oscillation, $f = 90 times / \min$ and $f = 95 times / \min$ ). The sampling period is 0.004 s; total sampling time is 3 s; sampling number is $N = 750$ . Based on equation (5), the identification value of total moment of inertia is $J = 0.1110 kg m^{2}$ .

The estimated value of load torque $T_{L}$ can be calculated with motor current according to equation (2) after the calculation of J and B. When $f = 90 times / \min$ , the estimated value of load torque $T_{L}$ is $5.389 \sin (3 π t) + 3.889 N m$ in sinusoidal oscillation and $T_{L}$ is $7.925 \sin (3 π t - A \sin (3 π t)) +$ $5.158 N m$ in non-sinusoidal oscillation. When $f = 120 times / \min$ , the estimated value of load torque $T_{L}$ is 6.4985 sin $(4 π t) + 5.1335 N m$ in sinusoidal oscillation, and $T_{L}$ is 12.2045 sin $(4 π t - A \sin (4 π t)) + 6.0845 N m$ in non-sinusoidal oscillation.

Figure 4.

Motor currents under two similar constant speeds: (a) $n_{r} = 460.305 r / \min$ $f = 90 times / \min$ and (b) $n_{r} = 485.877 r / \min$ $f = 95 times / \min$ .

Figure 5.

Motor currents under two similar time-varying speeds: (a) $n_{r} = 460.305 (1 - 0.4 \cos 3 π t) r / \min$ $f = 90 times / \min$ and (b) $n_{r} = 485.877 (1 - 0.4 \cos 19 π t / 6) r / \min$ $f = 95 times / \min$

The controllers of motor speed, q axis current and d axis current, are proportional–integral (PI) controllers (Siemens S120). The control type is $i_{d} = 0$ and the control relationships are

\begin{matrix} i_{q}^{*} = K_{p 1} (e_{1} + \frac{1}{τ_{1}} \int_{0}^{t} e_{1} dt) \\ U_{q}^{*} = K_{p 2} (e_{2} + \frac{1}{τ_{2}} \int_{0}^{t} e_{2} dt) \\ U_{d}^{*} = K_{p 3} (e_{3} + \frac{1}{τ_{3}} \int_{0}^{t} e_{3} dt) \end{matrix}

(7)

where $e_{1} = n_{r} - n$ ; $e_{2} = i_{qr} - i_{q}$ ; $e_{2} = i_{qr} - i_{q}$ ; $e_{1}$ , $e_{2}$ , and $e_{3}$ are the errors of speed, q axis current and d axis current, respectively; $K_{p 1}$ , $K_{p 2}$ , and $K_{p 3}$ are the proportional coefficients of speed, q axis current and d axis current controllers, respectively; $τ_{1}$ , $τ_{2}$ , and $τ_{3}$ are integration time constant of speed, q axis current and d axis current controllers, respectively; $i_{qr}$ and $i_{dr}$ are the given values of q axis current and d axis current, respectively; $i_{dr} = 0$ .

By parameter self-tuning of servo motor controller S120, the control parameters are obtained (Table 2).

Table 2.

Control parameters of servo motor controller S120.

Control parameters	Values	Units
$K_{p 1}$	0.408	$A / r / \min$
$τ_{1}$	43.2	$ms$
$K_{p 2}$	12.982	$V / A$
$τ_{2}$	2	$ms$
$K_{p 3}$	12.982	$V / A$
$τ_{3}$	2	$ms$

Nonlinear model of mechanical transmission parts

Mechanical transmission parts mainly include eccentric shaft and linkage oscillation platform. Models of mechanical transmission parts are established as follows.

Eccentric shaft and linkage structure are shown in Figure 6(a). O is the center of outer circle, and $O'$ is the center of inner circle. The simplified principle diagram is shown in Figure 6(b). The eccentric distance $OO'$ is $e = 9 mm$ and the linkage distance $OC = 573 mm$ .

Figure 6.

Eccentric shaft: (a) structural diagram of eccentric shaft and (b) principle diagram of eccentric shaft.

In Figure 6(b), the angle between eccentric distance $OO'$ and horizontal dotted line is $θ$ . Point O anticlockwise rotates with circle center $O'$ , and the radius is $OO'$ . Considering that OC is far longer than $OO'$ , the vertical displacement of Point C (namely, the vertical displacement of eccentric shaft $S_{e}$ ) is approximately equal to displacement of Point O. According to the projection relation, the vertical displacement of Point O is

S_{e} = e \sin θ

(8)

For the mold oscillation platform, the mold lies between Point C and Point D, and Point C performs approximately circular motion relative to the fixed Point D. According to the leverage principle, the displacement of mold is approximately proportional to the displacement of Point C. In Figure 1, the horizontal distance between Point C and Point D is about $1500 mm$ and the horizontal distance from the mold center to Point D is about $500 mm$ . Then, the vertical displacement of the mold is

S_{m} = k S_{e} = ke \sin θ = h \sin θ

(9)

where the approximate proportion coefficient is $k = (500 / 1500) = 1 / 3$ ; $h = ke = 3 mm$ is the amplitude of oscillation platform of continuous casting mold.

Nonlinear model of continuous casting mold platform

The system input is u (generally corresponding to $n_{r}$ ) and the output is $y = S_{m} = h \sin θ$ , indicating the nonlinear relationship. The mathematical model of continuous casting mold oscillation platform driven by servo motor is deduced on the basis of servo motor speed system model and mechanical transmission model. The structure diagram of the model is shown in Figure 7.

Figure 7.

Structure diagram of continuous casting mold vibration platform driven by servo motor.

Simulation experiments and error analyses

Simulation experiments

In order to verify the model, we adopted the same several typical input signals in the mathematical model and actual system, and analyzed the output curves.

First, we conducted relevant experiments with the oscillation platform of continuous casting mold in the laboratory. We developed the data acquisition program in the PC equipped with Siemens Step7 software, and used WinCC to inquire and monitor data. We acquired the mold displacement to PLC with displacement transducer and uploaded the data to PC for processing. Different motor speeds were set via the input terminal of actual system (the input signal is constant speed and time-varying speed). Then, we collected the data and obtained the curves of mold displacement at the output terminal.

Then, we used MATLAB/Simulink tools to conduct simulation. The same expected speed was adopted in the actual system and experimental model. The mold displacement was obtained with the model established in the article (simulation time was 3 s and the step size was 0.004 s).

The given displacement waveform function developed by DEMAG corporation is expressed as

S = h \sin ((ω t) - A \sin (ω t))

(10)

Figure 8 shows the output curves of the model and actual system under the constant speed (f = 90 times/min and $f = 120 times / \min$ , respectively). The waveform deviation rate $α$ is 0.

Figure 8.

Output curves of the model and actual system under the constant speed: (a) $f = 90 times / \min$ and (b) $f = 120 times / \min$ .

Figure 9 shows the output curves of model and actual system under the time-varying speed (f = 90 times/min and $f = 120 times / \min$ , respectively). The waveform deviation rate $α$ is 0.24.

Figure 9.

Output curves of the model and actual system under the time-varying speed: (a) $f = 90 times / \min$ and (b) $f = 120 times / \min$ .

As shown in Figure 8 and 9, the output curves of the simulation model and actual system are almost coincident with each other and show the better fitting results, which indicates that the established mathematical model in this article is close to the actual system.

Error analyses

The model error is discussed below. The error curves (Figures 10 and 11), respectively, corresponds to the curves in Figures 8 and 9. Relative error E is defined as

E = \frac{\sqrt{\sum_{1}^{N} e_{N}^{2} / N}}{\sqrt{\sum_{1}^{N} y_{N}^{2} / N}}

(11)

where $e_{N}$ is the error between model output and actual system output; $y_{N}$ is the model output value.

Figure 10.

Error curves under the constant speed: (a) $f = 90 times / \min$ and (b) $f = 120 times / \min$ .

Figure 11.

Error curves under the time-varying speed: (a) $f = 90 times / \min$ and (b) $f = 120 times / \min$ .

In the error curves, we can conclude that the model error is within 5%. The higher the frequency, the larger the error, because the higher frequency has the strong impact on the system response. The error analyses indicate that the established model in the article is reasonable and accurate.

Conclusion

With the oscillation platform of continuous casting mold driven by servo motor in the laboratory, an entire nonlinear system model, including servo motor speed system and mechanical transmission part, is established. Meanwhile, the viscous friction coefficient, moment of inertia, and load torque are identified.

Under the same input signal, the simulation output curves coincide with the actual output curves well, indicating the effectiveness of the established model.

The model error is less than 5%, indicating the effectiveness of the model.

Footnotes

Academic Editor: Liyuan Sheng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Item sponsored by the Natural Science Foundation of China and Baosteel Group Co., Ltd. (grant no. U1260203); Natural Science Foundation of Hebei Province (grant nos. F2015203400 and F2016203263); Doctor Foundation of Yanshan University (grant no. B960); the Cultivation Program Project for Leading Talent of innovation team in Colleges and universities of Hebei Province (grant no. LJRC013).

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Modeling and verification of the nonlinear system of oscillation platform of continuous casting mold driven by servo motor

Abstract

Keywords

Introduction

Structure of oscillation platform of continuous casting mold driven by servo motor

Modeling for oscillation platform of continuous casting mold driven by servo motor

Modeling of the servo motor system and parameter identification

Nonlinear model of mechanical transmission parts

Nonlinear model of continuous casting mold platform

Simulation experiments and error analyses

Simulation experiments

Error analyses

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References