GB2334691A - Skewed sinusoidal profile for displacement of a continuous casting mould - Google Patents

Abstract

A method of oscillating a mould, the method comprises applying an input to the mould to move the mould, the input defining a displacement profile with time, the profile being based on a sinusoidal profile, the sinusoidal profile being skewed. By this a system and method of mould oscillation is provided in which sticking is minimised and a practicable acceleration and velocity profile is employed.

Description

IMPROVEMENTS IN AND RELATING TO MOULD OSCILLATION This invention concerns improvements in and relating to mould oscillation, particularly, but not exclusively, for continuous casting of metal.

Continuous casting of metal involves the introduction of molten metal into the top part of a mould and the withdrawal of an at least partially solidified strand from the bottom part of the mould. To avoid sticking of the solidifying skin of the strand on the mould wall it is known to oscillate the mould in a reciprocating manner following the general direction of strand removal.

When the mould is travelling in the same direction as the strand it is desirable for the mould to be moving faster than the strand for quality reasons. This is so called"negative strip". In addition it is believed that many other benefits are conferred on the product if the time during which the mould motion opposes the direction of strand removal exceeds the time during which the mould motion coincides with strand removal.

However, the prior art systems often involve velocity profiles which vary unevenly and are undesirable for other reasons.

The present invention aims to provide a system and method of mould oscillation in which sticking is minimised and a practicable acceleration and velocity profile is employed.

According to a first aspect of the present invention we provide a method of oscillating a mould, the method comprising applying an input to the mould to move the mould, the input defining a displacement profile with time, the profile being based on a sinusoidal profile, the sinusoidal profile being skewed.

The method preferably oscillates the mould during casting of a strand. The oscillation preferably comprises motion concurrent with the direction of casting of the strand and motion in opposition to the direction of casting of the strand.

The profile may have an amplitude of between 1 and 20mm and more preferably of between 2 and 10mm.

The profile may have a frequency of between 20 and 600 cycles per minute, and more preferably of between 60 and 400 cycles per minute.

The profile preferably provides a ratio of time moving in opposition to the strand movement to time moving concurrently with the strand movement of between 1 : 1 and 4 : 1. Preferably a ratio of 1. 5 : 1 and 3. 8 : 1 and more preferably of between 2. 5 : 1 and 3 : 1. The ratio may be greater than 1. 5 : 1, or greater than 1. 7 : 1, or greater than 2 : 1, and may even be greater than 2. 8 : 1.

Preferably the asymmetry of the profile is determined by an asymmetry factor, z. Preferably z is between 0 and 1/A. w, where A is the amplitude of oscillation and w is the angular velocity. More preferably z is less than 0. 75/A. w. More preferably z is greater than 0. 1/A. w.

Preferably the profile is a product of a sinusoid and a skew matrix. The skew matrix may be defined as :-

1 Z Mskew = 0 1 where z is an asymmetry factor, preferably with the definition provided above.

The method may include the transfer of the input to the mould through mould support means. The input may be transferred to the mould by input applying means. The input applying means may be provided with the input by input defining means.

According to a second aspect of the present invention we provide apparatus for oscillating a mould, the apparatus comprising means for supporting the mould, means for applying an input to the support means and means for defining the input, the input defining a displacement profile with time for the mould, the profile being based on a sinusoidal profile, the sinusoidal profile being skewed.

The support means may comprise a frame for the mould.

The mould may be rigidly mounted on the frame or supported therefrom.

The input applying means may comprise one or more push rods. The push rods may define a configuration intended to multiply the amplitude of oscillation.

The input defining means may comprise one or more hydraulic actuators. The input defining means may incorporate one or more valves and/or one or more supply lines to provide and/or control the hydraulic actuation.

The input defining means may comprise a mechanical system. The input defining means may comprise one or more noncircular gears. For instance rotary input means may be connected, through a first non-circular gear to a second noncircular gear on output means. In this way the rotary input is converted to an output, suitable for a push rod, of the desired displacement profile. Preferably the first and second gears are non-circular and non-elliptical.

The profile defined by the input defining means may be provided according to the features, options and possibilities of the first aspect of the invention and/or as elsewhere defined in this document. various embodiments of the present invention will now be described, by way of example only, and with reference to the accompanying drawings, in which: Figure la illustrates a displacement against time plot for a prior art oscillator ; Figure lb illustrates a velocity against time plot for the prior art oscillator of Figure la ; Figure 2a illustrates a displacement against time plot for an oscillator employing a pair of elliptical gears ; Figure 2b illustrates a velocity against time plot for the oscillator of Figure 2a ; Figure 3 illustrates a displacement against time plot for an oscillator system ; Figure 4 illustrates a displacement against time plot for an oscillator system ; Figure 5a illustrates a displacement gainst time plot for an oscillator system ; Figure 5b illustrates a velocity against time plot for the oscillator of Figure 5a ; Figure 5c illustrates an acceleration against time plot for the oscillator system of Figure 5a and 5b ; Figure 6a illustrates a displacement against time plot for an oscillator according to one embodiment of the present invention ; Figure 6b illustrates a velocity against time plot for the oscillator of Figure 6a ; Figure 6c illustrates an acceleration against time plot for the oscillator of Figures 6a and 6b ; and Figure 7 illustrates parametric equations Y (t) and T (t) against parameter.

Molten material introduced into a mould during continuous casting cools within the mould to form a thin solidified skin.

By the time the strand exits the mould the skin is of sufficient thickness to contain the still molten core of the strand during subsequent transport and cooling.

The skin formed within the mould is relatively thin and fairly fragile as a result. Unless preventative steps are taken the skin tends to solidify on the mould wall and stick to it as a result. This can lead to ripping of the skin and other problems.

To counteract sticking it is known to oscillate a mould along the general path taken by the exiting strand. The type of sinusoidal oscillation employed is illustrated in Figure la.

The mould displacement illustrated has a cycle portion A during which the mould is moving upward relative to its centre starting position ; a maximum displacement upward, point B ; a downward movement portion C during which displacement decreases till the centre line D is reached ; a continuing downward displacement from the centre E ; a maximum downward displacement position F ; and a return cycle portion G during which the mould moves back up to the centre line.

The velocity profile for this displacement thus has correspond ng portions ; a decreasing upward velocity portion A ; a zero velocity transition point B ; a downward increasing velocity C ; a maximum downward velocity time D ; a decreasing downward velocity portion E ; a zero velocity transition F ; and an increasing upward velocity G.

If the mould velocity is compared with the downward speed of the strand, dashed line X, then for the time between T, and T2, the downward velocity is greater than the strand downward velocity (670 of the downward cycle in this case). Such sinusoidal motion is generally produced by an eccentric shaft and push rod arrangement, though it may be by hydraulic actuation.

It has been proposed to increase the proportion of the downward cycle in which the strand velocity is exceeded, for instance by the use of equivalent, but 90 out of phase elliptical gears.

A displacement profile and velocity profile for such a system is illustrated in Figures 2a and 2b respectively.

The mould displacement illustrated has a cycle portion A during which the mould is moving upward relative to its centre starting position ; a maximum displacement upward, point B ; a downward movement portion C during which displacement decreases till the centre line D is reached ; a continuing downward displacement portion E ; a maximum downward displacement position F ; and a return cycle portion G during which the mould moves back up to the centre line.

The velocity profile for this displacement thus has a relatively steady upward velocity portion A1 ; a decreasing velocity portion A2 ; a zero velocity transition point B ; a downward increasing velocity Cl ; a steady downward velocity portion C2 ; the maximum downward velocity time D ; a steady downward velocity E1 ; a decreasing downward velocity portion E2 ; a zero velocity transition F ; and an increasing upward velocity G1, leading back to a steady upward velocity portion G2.

If the velocity is compared with the downward speed of the strand, dashed line X, then for the time between TX and T2, the downward velocity is greater than the strand downward velocity (910 of the downward cycle in this case).

Such techniques, based around a symmetrical displacement and symmetrical velocity profile, still face problems from the kinematics required for their operation.

Velocity profiles, such as those of Figures 3 and 4, employ asymmetric profiles to an extent.

The profile of Figure 3 generally follows the path of a saw tooth with a slower upward velocity, portion N, than downward velocity, portion M, used. The change from linear velocity in one direction to linear velocity in the other direction cannot in practice give a true saw tooth form as the infinite acceleration this calls for is not possible. The net result, therefore, is a rounding, P, of the velocity variation based on the greatest acceleration the system can practically provide. This profile introduces significant problems into the system as large accelerations, and hence forces, are needed to give a reasonable approximation to the profile desired.

However, the actual profiles achieved are unpredictable, arbitrary and variable. For this reason process modelling is impracticable, eg, calculating negative strip time.

The system of Figure 4 also fails to work fully. Here the upward part of the velocity is defined by a sinusoidal profile of first frequency, portion Fi, and the downward part of the velocity is defined by a sinusoidal profile of greater frequency, portion F2. Whilst the velocity profile this gives is continuous the corresponding acceleration profile is not.

As a consequence the required profile calls for infinite acceleration during the transition. This level of force is not available and as a result a period of"readjustment"is needed at the top and bottom of the stroke. The result is that undesirable anomalies in the displacement curve are created.

Additionally the exceptionally high level of jerk introduced is mechanically undesirable and can lead to damage of the fragile strand skin.

Figure 5a shows a profile obtained by superimposing a sinusoidal variation onto the normal constant input angular velocity described in relation to Figure 1. This input is usually provided by hydraulic means. The result is an output at the mould which descends more quickly than it rises. There are, however, limits to the asymmetry possible using this method. As Figure 5b shows, if the asymmetry is too great, for instance, time up : to time down is greater than 1. 5 : 1, then a mid stroke reduction in velocity occurs with undesirable results. Further increases in asymmetry, 2. 8 : 1, cause the direction of mould travel to reverse during the upward stroke.

The optimum solution set out in Figures 6a, 6b and 6c overcomes all of these problems in providing a system which minimises stick, is mechanically practicable and reliable and which avoids skin damage through sudden changes. The continuous nature of the skewed, and hence asymmetric, sine wave overcomes many of the problems faced with the prior art and other solutions discussed above.

The system is defined by a displacement-time curve which is a function of a sinusoid and a skew matrix.

Sinusoidal mould displacement, relative to the mid stroke position, w. r. t. time is defined by : y (t) = A. sin (8 (t) where A is the amplitude of oscillation and 9 is the angular orientation relative to the mid stroke position.

For constant angular velocity (w) this reduces to : y (t) = A. sin (M. t) Consider transformation of t & y. If t maps to t + zy, and y to y then in matrix form this case can be represented by :

T (t) 1 z t Y (t) 0 1 y (t) Where

1 z 0 1 is the skew matrix Plotting parametric equations Y (t) against T (t) with parameter t ranging between r/2 and 7/2 gives the profile of Figure 7.

From Figure 7 the difference between upward and downward motion can readily be appreciated.

For the purpose of further manipulation the relationship Y (T) is required. In effect Y (t (T)) is required, but T (t) not t (T) is known.

That is T (t) = t + z x A. sin (w. t) Since a symbolic result for t in terms of T cannot be derived, a polynomial relationship between t and T is assumed : Hence t (T) = S0.T+S1.T3+S2.T5+S3.T7-S4.T9-S5.T11 Using regression (such as Mathcads'Linfit') to evaluate the coefficients in conjunction with particular values of A, w and z gives : ST = (0. 6895, 2. 2557, 71. 6990,-VE2300, 56080,-VE424319) The displacement is therefore given by : Y (T) = A. sin [@ (SoT + S, T' + S2T5 + S3T7 + S4T9 + S5T11) ] In reality it is the angular velocity not time which fluctuates, however the effect is the same.

By differentiation the velocity and acceleration functions are also derived : v (T) = A. cos [0) (SoT + S1T3 + S2T5 + S3T7 +S4T3 + S5T11)] #. (S0 + 3S, T2 + 5. S2T4 + 7. S3T6 + 9.S4T8 + 11.S5T10) a (T) = A. sin [@ (SoT + S1T3 + S2T5 + S3T7 + S4T9 + S5T11)]#2. (S0 + 3. S1T2 + 5. S2T4 + 7. S3T6 + 9. S4T8 + 11.S5T10)2 + A. cos [1 (SoT + S1T3 + S, tus + S3T7 + S4T9 + SsT"-) Ico. (6. ST +20. S2T3 + 42. S3T3 + 72.S4T7 + 110.S5T9) If z is now set to 9 the skewed displacement profile is as illustrated in Figure 6a. The displaced profile sequence commences with a portion A in which the mould moving upwards away from the centre line ; a transition point B of maximum upward displacement ; a portion of downward movement C ; a point D at which the mould passes the centre line ; a continued downward movement portion E ; a maximum downward displacement transition point F ; and an upward movement portion G back to the centre line. The sequence is then repeated.

The profile includes a greater portion of time in which the mould is being displaced upwards than in which the mould is being displaced downwards, as is clear from the velocity profile of Figure 6b.

The profile has a decreasing upward velocity portion A which gradually reduces ; a zero velocity point B ; an increasing downward velocity C ; a maximum downward velocity point D ; a decreasing upward velocity portion E ; a zero velocity point F ; and an increasing upward velocity portion G.

The cycle is made up of approximately 65t upward movement time and approximately 35% downward movement time. Thus only for a substantially reduced, and hence relatively small, part of the cycle is the mould moving concurrently with the strand.

Furthermore, within that downward part of the cycle the downward velocity of the mould exceeds the strand for the substantial part of it. As shown in Figure 6b between T, and T2 the downward velocity X of the strand is exceeded, with this portion equating to two thirds of the downward part of the cycle.

This profile is achieved, as shown in Figure 6c, using an acceleration profile which is both theoretically possible and which in practical terms does not highly load the oscillating system. A more reliable and longer lasting system results.

The determined profile can be implemented through a number of mechanical systems. The mould and support framework can be oscillated using hydraulic cylinders the input to which is controlled to give the desired displacement and hence velocity profile. Equally the profile can be implemented using gears profiled to convert the constant angular velocity input of the drive to the necessary variable angular velocity at the eccentric.

Claims (12)

1. CLAIMS : 1. A method of oscillating a mould, the method comprising applying an input to the mould to move the mould, the input defining a displacement profile with time, the profile being based on a sinusoidal profile, the sinusoidal profile being skewed.
2. 2. A method according to claim 1 in which the profile provides a ratio of time moving in opposition to the strand movement to time moving concurrently with the strand movement of between 1 : 1 and 4 : 1.
3. 3. A method according to claim 2 in which a ratio of between 2. 5 : 1 and 3 : 1 is provided.
4. 4. A method according to claim 1 in which the asymmetry of the profile is determined by an asymmetry factor, z, z being between 0 and 1/A. w, where A is the amplitude of oscillation and w is the angular velocity.
5. 5. A method according to claim 4 in which z is less than 0. 75/A. w.
6. 6. A method according to claim 4 in which z is greater than 0. 1/A. w.
7. 7. A method according to claim 1 in which the profile is a product of a sinusoid and a skew matrix.
8. 8. A method according to claim 7 in which the skew matrix is defined as :

   Z Mskew = to l where z is an asymmetry factor.
9. 9. Apparatus for oscillating a mould, the apparatus comprising means for supporting the mould, means for applying an input to the support means and means for defining the input, the input defining a displacement profile with time for the mould, the profile being based on a sinusoidal profile, the sinusoidal profile being skewed.
10. 10. Apparatus according to claim 9 in which the input applying means comprise one or more push rods, the push rods defining a configuration intended to multiply the amplitude of oscillation.
11. 11. Apparatus according to claim 9 in which the input defining means comprise one or more hydraulic actuators.
12. 12. Apparatus according to claim 9 in which the input defining means comprise one or more non-circular gears.