Continuous Steel Casting System Components

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1 CCC Annual Reort UIUC, August 9, 5 Caturing and Suressing Resonance in Steel Casting Mold Oscillation Systems Oyuna Angatkina Vivek Natarjan Zhelin Chen Deartment of Mechanical Science & Engineering University of Illinois at Urbana-Chamaign Continuous Steel Casting System Comonents Mold oscillation adle Casting direction Sticker Tundish Breakout Mold Water cooling srays Suorting rolls From CCC University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen

2 Primary Beam Position of hydraulic iston (not in icture) under the beam Production Unit Resonance Problem and Project Objective Pivot Mold Table Problem:. The rimary beam main resonance mode starts exhibiting excitation when the reference frequency aroaches one-third of the resonance frequency. The mold dislacement and velocity rofiles distortion is found to be mainly caused by this resonance 3. The reason for the onset of beam resonance excitation has not been identified 4. Distortion has not been removed, oeration in the desired frequency range has not been attained Secific Project Objective: Model this mold oscillation system, simulate it, identify what causes excitation of the rimary beam resonance, and eliminate the distortion University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 3 Position and Velocity Profile from Plant In going to higher frequencies, while reducing the oscillation amlitude, the velocity rofile is found to become highly distorted. Position rofile Velocity rofile (time derivative of osition rofile) University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 4

3 Hardware Testbed: Simlified Instrumented ayout Sensor of vertical mold dislacement (x m ) Beam Hydraulic actuator M O D Sensor of actual iston osition (x ) Hardware testbed catures similar resonance roblem ) Resonance frequency of beam 9. ) Reference inut to the iston for tracking - sinusoid at 4.6 3) Mold osition rofile is highly distorted Hydraulic valve/actuator Nonlinear behavior (same model as lant) Note: Although the objective is a distortion-free mold velocity rofile, we focus on iston and mold osition rofiles observed through sensor signals, since a distortion-free (ure sinusoidal) dislacement guarantees a distortion-free velocity. University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 5 Hardware Testbed Validation: Exerimental Data Exhibits Resonance Problem Piston osition (reference am=3 mm, freq=4.6 ) Mold osition The desired osition for the iston reference inut - is a sine wave of frequency f r =4.6 (half of 9. - the resonance frequency of the beam) and amlitude 3 mm P controller with gain K= is used Piston osition signal looks almost erfect But large distortion (deviation from sinusoidal rofile) at the mold end is observed This haens when the frequency of the iston osition reference is near a submultile (exact integer fraction) of the beam resonance frequency University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 6

4 Software testbed: beam model: set of Timoshenko beam (linear) PDEs with boundary conditions E = Ga Youngs modulus G = 8 GPa Shear Modulus for steel ρ I x = l y x = l x x ( ) t b eft beam x = Right beam Mg (Mold weight) y b γ y ' y + y = b ψ b yr yr yr m k Ga m g m b γ ' y k Ga + = b ψ R mb g t t x x t t x x D dynamic beam bending angle PDE 3 =787 Kg / m - density of steel a b.88 m cross section area of beam 5 4 =. m Moment of inertia of beam mb = 69 Kg / m Mass er unit length of beam M=5 Kgs mold mass k ' =.83 Shear constant Beam width = 5.3'(hollow with thickness.94') l = 34.5 ' Beam breadth = 6' (hollow with thickness.38') Daming coefficients γ γ γ Kg / sec R conditions D dynamic beam transverse dislacement PDE ψ ψ ψ y b + γ ' ψ = + b ψ I m EI k Ga a t t x x x Boundary conditions at x=-l y ( l) = x ( t) ψ ( l) EI = x Boundary conditions at x= y () = y () = ψ () = ψ () R R ψ R () ψ () EI = EI x x = = = = Testbed arameters University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 7 b Model of two beams couled at the center via boundary conditions Mold dynamics is one of the boundary ψ R ψ R ψ R yr b + γ ' ψ = + b ψ R I m EI k Ga a t t x x x Boundary conditions at x=l ψ R ( l) EI = x y R ( l ) y R ( l ) y R ( l ) k ' Gab ψ R ( l ) + Mg + M + γ m = x t t y = linear dislacement (right side of the beam) R ψ = angular dislacement (right side of the beam) R y = linear dislacement (left side of the fulcrum) ψ = angular dislacement (left side of the beam) PDE variables Software testbed: hydraulic servo model: set of nonlinear ODEs and their couling to beam PDEs A Piston dynamics showing the couling with the beam The osition of the iston is a boundary condition for the beam equations and the shear force in the beam acts on the iston, couling the two models together B ( l) y mɺɺ x + bxɺ = ( PA PB ) a mg + k ' Gab ψ ( l) x University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 8 F Turbulent Flow Equations for flow in chambers A and B c ( d xs ) Ps PA xs < d qa = c ( d xs ) Ps PA c ( xs + d ) PA Pt d < xs < d c ( xs + d ) PA Pt xs > d c ( d xs ) PB Pt xs < d qb = c( d xs ) PB Pt c( xs + d ) Ps PB d < xs < d c( xs + d ) Ps PB xs > d x sool osition - ositive when the sool moves to the right. s Its mean osition is with valve underla ga of d on both sides. ω, ζ - sool filter arameters, s A s q flow rate for chamber A - ositive when oil flows in to A q B α flow rate for chamber B - ositive when oil flows out of B A,B,s,t Chamber connected to A and B, source, tank Pressure equations ( P and P - ressure in A and B, resectively Pɺ β = q a x ( ( )) VA + a + x ɺ ) A A Pɺ β = q + a x ( ( )) VB + a x ɺ ) B B x iston osition - ositive if it moves to right of midoint of cylinder V, V static volume of chambers A and B A B ( ) A a + x dynamic volume of chambers A and B half the iston stroke length a surface area of iston Electronic control of sool osition ɺɺ x xɺ x u s + ζ sωs s + ωs s = ωs. Fast dynamics - hence not modelled in simulations B

5 Software Testbed: Numerical Model MATAB Simulink Diagram 3. Finite-difference aroximation (ODE) of Timoshenko beam equation (PDE). Nonlinear iston hydraulics 3. Control algorithm (to be designed) University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 9 beam valve Software Testbed: Numerical Model PDE Aroximation Finite difference aroximation satial derivatives in PDE are aroximated by difference equations f f ( x + x) f ( x x) f f ( x + x) f ( x) + f ( x x) x x x ( x) Result is a set of couled time-odes imlemented in state-sace form. Nonlinear servo model ODE is imlemented in state-sace form 3 control Controller is imlemented in state-sace form University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen

6 Software Testbed (Numerical Code) Validation Software (Matlab) rogram was written that comutes analytical model resonse to inuts using numerical algorithms Parameters in the beam model were chosen to obtain a resonance frequency at 9. Reference inut to the iston for tracking sinusoid at 4.6 Proortional controller used with gain of.6 Mold osition exhibits distortions similar to those of the testbed Therefore, numerical model simulator can be used as a latform for understanding the testbed dynamics and for testing controllers We ve got a tool for in silico exerimentation - our own software testbed to lay with! Mold osition University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen Piston osition (reference am=3 mm, freq=4.6 ) Alication of Mold Oscillation Model to Severstal Caster For Severstal caster, arameters needed for the model can t be directly measured due to comlexity of assembly, which is similar to Nucor. Change model arameters for 9. resonance at Nucor Decatur mold oscillation system to Severstal with an measured 5. resonance frequency Retune k factor to adjust cross-sectional moment of inertia by factor of k. k is the value that maximizes the mold dislacement magnitude resonse at the desired resonance. For Severstal, k is found to be.483, yielding damed natural frequency 5.4 and resonance frequency 5.5. Adjustment of each individual model for runtime (dicretization accuracy) versus resonance frequency to allow model to run in real time while still maintaining the required resonance. University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen

7 Resonance frequency analysis Set solution of two couled beams equation as (, ) ( ), ψ (, ) ψ ( ), ( ) j ω t j ω t j t ω y x t = y x e x t = x e x t = x e Then, the corresonding equations for both the left and right beams have the form γ y k Ga b y ψ γ ab Eab k Gab y y j y g, j ψ ψ ω + ω = ω ψ + ω ψ = + ψ mb mb x x Imb mb x Imb x or in matrix form k Gab γ y k Ga b r + ω jω r mb mb mb y ( x ). = k Ga ( x ) b Eab γ ψ ab k Ga ψ b r r + ω jω Imb mb Imb Imb Equating determinant to zero yields equation below with four solutions: ±r and ±r 4 mb γ ψ γ y mb 4 3 γ ψ ab γ y r + r ω + jω + + ( ω jω + ab E k G EI k Gab k Gab Imb mb γ γ a γ k Ga y ψ b k Gab y b ω + j + ω = Imb Imb Imb University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 3. General solution ( ) ( x ) Then, the general solution is y x y rx yɶ yɶ yɶ yɶ = C e = Cɶ e + Cɶ e + Cɶ e + Cɶ e ψ ψ ψ ψ ψ r x r 3 4 x r x r x 3 4 ɶ ɶ ψɶ 3 ɶ 4 with eigenvalues y k ɶ b i b i Ga r m ψ = i ( k Gabri ( mbω ɶ + jωγ y )) mb Now we substitute this solution into boundary conditions below: ( ) ψ ( ) ( ) R ( ) ( ) ( ) EI ( ) EI ( ) EI ( l ) y l =, EI l =, y =, y =, ψ = ψ, ψ = ψ, ψ =, R R R 4M ω + γ m k Gab ( y R ( l ) ψ R ( l )) ψ R ( l ) =. 4M, University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 4

8 Frequency resonse calculation r l r l r l r l = C y e C y e C y e C y e x r l r l r l r l ψ ψ C r e r C e + r C y e r C y e =, C y + C y + C y + C y =, C y + C y + C y + C y =, R R 3R 3 4 R 4 C y + C y + C y + C y = C y + C y + C y + C y C This substitution (of the general solution into the boundary conditions) yields rψ - C rψ + C r ψ - C r ψ = C rψ - C r ψ + C r ψ - C r ψ, R R 3R 3 4 R 4 r l -r l r l -r l r l -r l ψ - ψ + 3 ψ 3-4 ψ 4 = R ψ - R ψ C r e C r e C r e C r e C r e C r e r l -r l 3R ψ 3 4 R ψ 4 + C r e - C r e, r l -r l r l -r l R R 3R 3 4 R C e D + C e D + C e D + C e D 4 =, ( ψ ) ( ω ωγ m ), b ( ψ ) ( ω ωγ m ) ( ψ ) ( ω ωγ m ), b ( ψ ) ( ω ωγ m ) D = k Ga r y y M + j b D = k Ga r y y M + j D = k Ga r y y M + j 3 b D = k Ga r y y M + j ,,, Factoring out all the coefficients C obtain matrix equation. Using tabled arameters solve for roots r. University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 5 Software testbed: frequency resonse x = l y x = l x x ( ) t eft beam x = Right beam Mg (Mold weight) the frequency resonse can then be obtained from the equation for the mold vertical dislacement at the right end: ( ) = ɶ + ɶ + ɶ + ɶ r l r l r l r l yr l CR ye C R ye C3R y3e C4 R y4e.e+.e- ( f r =9.4,.8) Amlitude.E-.E-3.E-4 ( f r =695.57,.9) ( f r =94.3,.9).E-5.E-6.E Frequency (in ) University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 6

9 Numerical modeling Mold oscillation testbed is modeled by the second-order finite difference scheme ( ( ) ( ) ( )) ( ψ ( ) ψ ( )) y t = B y x x y x + y x + x + B x x x + x D y t g The zero moment boundary condition for the left end is ( l ) ( l x ) ( l x ) ψ ψ + ψ EI EI =, x x ψ, ( l x ) ψ ( l + x ) ( l ) ψ ( l + x ). ψ Same method can be alied at the right end. For the boundary condition due to mold reaction force: ( ) γ ( ) k Ga ( ) ( ) yr l m yr l b yr l yr l x k Gab + ψ R ( l x ) g t M t M x M For the boundary condition of equal moments at the hinge (Taylor exansion) Since ( x ) ( ) x ( ) x ( ) ( x ) ( ) x ( ) x ( ) ψ = ψ + ψ + ψ +, ψ = ψ + ψ + ψ +, ψ ( ) = ψ ( ) R ( x ) the angular dislacement is ψ ψ R ψ, R ( ) + ( ψ ( x ) + ψ R ( x )) ( x ) University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 7 Mold Oscillation System Parameters Parameter Variable Nucor-Steel Nominal value Severstal Units Mass of beam er unit length m b kg/m Area of cross section of beam a b.88.4 m Shear modulus G Pa Modulus of elasticity E Pa Moment of inertia of the beam cross-section I m γ Beam transverse dislacement y kg/(m sec) Beam angular dislacement γ ψ kg/(m sec) Mold daming γ m kg/(m sec) Shae of the cross-section k factor Mold mass 5 5 kg M l Half of beam length m University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 8

10 Numerical Model Stability The maximum real arts of eigenvalues vs. discretization nodes number: X = AX + Bu, t Y = CX + Du. The maximum real arts of eigenvalues Unstable Stable When the real art is negative, the system is stable For most discretization nodes number, the system is stable University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 9 Software Testbed: Resonance Frequency Deendence on Satial Discretization Ste Size x Damed Nat. Nucor-Steel Resonance Damed Nat. Severstal Resonance l/ l/ l/ l/ l/ l/ l/ Analytical solution Normalized frequency is ratio of simulated resonance and analytical solution University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen

11 Runtime Deendence on Satial Discretization Ste Size x Damed Nat. Nucor-Steel Resonance Damed Nat. Severstal Resonance l/ l/ l/ l/ l/ l/ l/ Analytical solution University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen Real-time Virtual Testbed: Parameter Matching for the Desired Resonance Frequency and Runtime Actual arameters setting Checking at what n matrix A is Hurwitz x Choosing number of nodes for model x Mold mass, kg Nucor-Steel Resonance Mold mass, kg Severstal Resonance l/ l / l / l / l / l / l / if unstable Finding mold mass to match natural frequency Checking convergence of numerical model University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen

12 - Hardware testbed erformance diagnostics The initial conjecture: nonlinear actuator dynamics roduces harmonics of 4.6 distorting the iston osition, which in turn distorts the mold osition Piston osition (reference am=3 mm, freq=4.6 ) Mold osition University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen sectrum of iston osition sectrum of mold osition The mold osition is distorted, but the iston osition looks erfect to the naked eye both in time domain and frequency domain (no visible harmonics). Since the actuator is visually observed to erform ideally, the initial conjecture looks wrong Catching the culrit with a magnifying glass! sectrum of iston osition near the resonance frequency. sectrum of mold osition near the resonance frequency However, zooming into the magnitude sectrum of the iston osition reveals a small eak at 9. Naked eye is easily misled! Missed small iston osition distortion due to nonlinear actuator dynamics Peak is small, but critical - at resonance frequency it is amlified by about 4 times to cause a large eak at the mold end! This matches with other exeriments, in which a sinusoidal iston osition amlitude at 9. is amlified by the beam by a factor of 3 at the mold end Conclusion: The small eak in iston osition being at resonance frequency excites the beam resonance and creates the large eak in the mold osition. All revious attemts at solving this roblem failed because the distortions were thought to start in the beam. In fact, however, they start in the iston, and are just amlified by the beam. University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 4

13 Source of Distortion (Revisited) Piston osition (reference am=3 mm, freq=4.6 ) Mold osition Simulation result: iston and mold osition with iston reference at 4.6 sectrum of iston osition near the resonance frequency sectrum of mold osition near the resonance frequency Simulation result: magnitude sectra of iston and mold osition near resonance frequency Piston osition looks erfect but has small eak at 9. in magnitude sectrum generated by the nonlinear servo model This small eak is amlified by beam causing significant distortion in the mold osition Software testbed generates the resonance roblem and can be used for initial testing of otential solutions (modeling is NOT necessary for the solution aroach develoed) University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 5 Solution Technique oo Augmentation r + - K P x x reresents the dislacement of iston x tracks reference sinusoid r of frequency f r well but has a small undesired sinusoidal comonent at frequency nf r r + + K P - + q F Additional loo x augment closed loo with controller F as shown then x continues to track r well, but has no undesired sinusoid at frequency nf r University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 6

14 Software Testbed: Simulation Results with the Filter-Nucor Controller develoed is alied to the model Internal model controller takes the form of filter F with ζ=. and ω r = πf r =π X 9. rad/sec Controller K=.6 and reference sinusoid amlitude 3 mm and frequency f r = 4.6 Distortions in mold osition rofile comletely eliminated - Piston osition (reference am=3 mm freq=4.6 ) Mold osition sectrum of iston osition near the resonance frequency sectrum of mold osition near the resonance frequency University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 7 Digital Controller Imlementation: Hardware Testbed-Nucor Controller develoed is alied to hardware testbed to remove harmonic at 9. from the iston osition Filter F introduced with ζ =. and ω r =π X 9. rad/sec Controller K=, reference sinusoid amlitude 3 mm and frequency Piston osition (reference am=3 mm, freq=4.6 ) Mold osition sectrum of iston osition near the resonance frequency sectrum of mold osition near the resonance frequency University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 8

15 Mold Position Distortion of Severstal Model Simulation result: mold osition with iston reference at.5 Inut at the frequency.5 leads to the distortion in the mold dislacement rofile due to small amlitude sinusoid at the resonance frequency of the Severstal lant in the inut from the actuator University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 9 Alication of the Control aw to Severstal Model Simulation result: mold osition with iston reference at.5 the resonance magnitude reduced by 4 times and the mold osition distortion is eliminated University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 3

16 Effect of Project Results on Casting Product Quality Casting at a lower frequency of oscillation for the mold enforced by simle roortional controllers Casting at a higher frequency of oscillation for the mold enabled by the roosed controller - shallower surface marks and hence less likely to turn into surface cracks University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 3 Conclusions (Past) The mold velocity distortion roblem at Nucor Steel, Decatur has been solved for a hinged beam-tye mold oscillation system by imlementing a new feedback control law based on the disturbance model. Software testbed heled! The new control law eliminates eriodic mold velocity disturbances, such as those caused by excitation of system natural frequencies by actuator nonlinearity, without the need for system model This imroves the surface quality of the steel being roduced and also enables the roduction of other crack sensitive grades of steel The controller is now a ermanent feature of caster oeration University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 3

17 Conclusions (New) An adjustable software simulation testbed has been develoed that ermits recreating the distortion roblem in both thin and thick continuous casters This testbed retains the exact resonance frequency matching for a broad range of comuting caabilities through simle one-arameter adjustment As a result, the software testbed develoed can be run in real time and connected to the roduction control system for mold oscillation controller testing and debugging, without the need for building the hardware testbed University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 33 Acknowledgments Continuous Casting Consortium Members (ABB, AK Steel, ArcelorMittal, Baosteel, JFE Steel Cor., Magnesita Refractories, Nion Steel and Sumitomo Metal Cor., Nucor Steel, Postech/ Posco, SSAB, ANSYS/ Fluent) Engineers in Nucor Steel Decatur who worked on mold oscillation roject. Dr. Petrus, (former grad student, currently at Nucor) for heling with Simulink rogram and advice. Prof. Thomas and Prof. Bentsman for their atience, knowledge and guidance on this roject. University of Illinois at Urbana-Chamaign Metals Processing Simulation ab Zhelin Chen 34