Modeling of Mold Oscillation. Mold Oscillation System at NUCOR

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1 NNU REPORT 8 UIUC, gst 6, 8 Modeling of Mold Oscillation Vivek Natarajan (PhD Stdent) Joseph entsman rian G. Thomas Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Mold Oscillation System at NUCOR Mold displacement and velocity Mold Table Pivot Primary eam Position of Hydralic ctator (not in pictre) nder the beam Problem: 1. Resonance mode of primary beam gets ecited when the actator oscillates at one-third the resonance freqency. This nwanted resonance distorts the mold displacement and velocity profile Objective: Model this mold oscillation system, simlate it, identify the sorce of distrbance and control it University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas

2 Simplified mock-p eam M O D Hydralic ctator Mock-p captres similar resonance problem 1) resonant freqency = 9.6Hz ) inpt at 4.8Hz ecites 9.6Hz eam is modeled sing Timoshenko beam model Heavy Mold, significant dynamics acconted for in ondary conditions Hydralic actator Nonlinear behavior (same model as plant) eam and actator copled sing bondary condition University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 3 Timoshenko eam Model wt () eft beam Right beam = Mg (Mold weight) 1D dynamic beam transverse displacement PDE k ' G y ψ y y mg γ m = t t = l y = l R R R k ' G y ψ y y R mg γ R m = t t 1D dynamic beam bending angle PDE y EI ψ k ' G ψ γ ψ k m ψ R yr R R + = EI ψ k ' G + ψ R γ ψ R k m ψ = t t t t ondary conditions ψ ( l) ψ R () y ( l) = w( t) EI = yr() = EI = M ψ R() = S ψ () yr() l yr() l ψ R() l y() = EI = M ψ () = S k ' G ψ R( l) + Mg + My R( l) + γ = EI = t E = Gpa Yongs modls G = 8 GPa Shear Modls for steel ρ I Mock-p parameters 3 =787 Kg / m - density of steel rea =.88 m cross section area of beam 5 4 =. 1 m Moment of inertia of beam m = 69 Kg / m Mass per nit length of beam M=5 Kgs k ' =.83 Shear constant eam width = 5.13'(hollow with thickness.94') l = 34.5' eam breadth = 6' (hollow with thickness.38') Solve these 4 Partial differential eqations simltaneosly University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 4

3 Simlated displacement histories 1-3 Free vibration de to gravity.1 Forced vibration (amp-.1m, freq-9.6hz) Displacement of mold in meters Displacement of mold in meters Simlation of the for partial differential eqations with the actal parameters indicates the beam resonance freqency is 1.6 Hz Simlation after changing moment of inertia slightly (factor of 1.15) gives a resonance freqency of 9.6Hz Eperimental resonance freqency is 9.6Hz University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 5 Hydralic ctator Electronic control of spool position Fast dynamics - Hence not modelled in simlations ctator dynamic eqations mw = ( P P) + F ( fric) w w - ctator displacement Set of Nonlinear Differential Eqations F Trblent Flow Eqations for flow in and c( d s) Ps P s q = c( d s) Ps P c( s + d) P Pt d < s c( s + d) P Pt s c( d s) P Pt s q = c( d s) P Pt c( s + d) Ps P d < s c( s + d) Ps P s is positive when the spool moves to the right. Its mean Srface area of piston University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 6 s position is with valve nderlap gap of d on both sides. q q is positive when oil flows in to chamber is positive when oil flows ot of chamber qp, Volme flow rate and Pressre α Chamber connected to and, sorce, tank,,s,t Pressre eqations P = q w V + + w P = q + w V + w w positive if it moves to right of midpoint of cylinder V, V Static volme of chambers and ( + w) dynamic volme of chamber and Half the piston stroke length

4 Hydralic ctator contd. Force acting on actator de to beam F y = kg ψ = l Control law that determines spool position Control law (, ) ( ) ( ) = f w w = k w w + k w w dt s des p des I des t State space form of actator eqations = 1 ( ) ( ) = fric kg m = l V = q 3 V + 1 = q + 4 ( ) 3 ( ) ( ) ( ) ( ) 4 ( ) ( ) ( ) 4 ψ c d s Ps s q = c d s Ps 3 c s + d 3 Pt d < s c s + d 3 Pt s c d s Pt s q = c d s 4 Pt c s + d Ps 4 d < s c s + d Ps s = s Parameters for mock-p 6 d = m =.46m P = 3Psi s P = 3Psi t c = V =.875inch 3 V = 4.3inch m = 8Kgs = 15Mpa =.15m fric = 1 Ns. / m University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 7 Flow diagram for copled actator-beam control simlation P In1 q Ot 1 In Flowrate from pressre In 1 P In Ot1 In 3 Pressre dynamics In1 Ot 1 In q In1 P In Ot1 Flowrate from pressre In3 Pressre dynamics Solve with Matlab Simlink In 1 Ot1 In Ot In 3 ctator dynamics dot Ot1 In 1 In Ot1 Reference signal for actator PI Controller ctator Controller 1 Constant 1 ' = + y = C+D State -Space f() Shear _stress eam University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 8

5 Possible cases of problem When actator is given a sinsoidal reference inpt at a freqency half the resonance freqency, the resonance freqency of beam gets ecited. Nonlinear pressreflow eqations generate resonance freqency? Nonlinearities at bearing generate distrbance torqe? eam ehibits nonlinear behavior (probably from the heavy mold mass)? Investigate with model simlation University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 9 Mockp Simlation with well-tned controller: Is nonlinear pressre-flow eqation the case of problem? Reference inpt to actator in meters Mold displacement in meters Reference inpt to actator at half the resonance freqency Mold displacement at half the resonance freqency ctator displacement in meters Mold velocity in meters per second ctator displacement at half the resonance freqency Mold velocity at half the resonance freqency Carefl tning of PI controller (k P =1.6, k I =.) ensres that the actator otpt has small second harmonic content Small distortions ehibited by mold displacement and velocity profiles Distortions less than those noticed in eperiments Nonlinear pressre-flow eqations may not be the sorce of troble if controller is properly tned. University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 1

6 Other sorces of resonance freqency Simlation indicates bearing friction torqe is probably not the sorce of distrbance. Other types of bearing vibrations have not yet been considered. In case beam dynamics is identified to be the sorce of problem, a more complicated higher dimension beam model will be considered. Earlier reports on this problem by other grops indicate that the actator is the sorce of resonance harmonic and the slow spool pdate rate cold be a reason for this. Hence other nmodeled dynamics and nonlinearities not inherent in the actator need to be considered University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 11 Mockp Simlation with delay in spool inpt Mold displacement in meters Reference inpt to actator in meters Reference inpt to actator at half the resonance freqency Mold displacement ctator displacement in meters ctator displacement at half the resonance freqency With a delay of.5s introdced between the controller and the actator, the actator displacement starts ehibiting higher freqencies at half the resonance freqency Mold velocity at half the resonance freqency These freqencies give.4 mild distortions in the mold displacement and. get magnified in the velocity Hence similar nmodeled dynamics in -. the actator cold be the sorce of higher harmonics Mold velocity in meters per second University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 1

7 Conclsions and ftre work Simlations sggest that nonlinear pressre-flow behavior of the actator does not eplain the resonance freqency. dditional nonlinearities and nmodeled dynamics sch as delay might be responsible. Ftre eperiments will be performed sing the mockp to qantify these nonlinearities by measring actator pressre, & displacement and velocity at varios points in the beam Improving controller design depends on the sorce of the distrbance. University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 13 cknowledgements Ncor Steel Decatr - Ronald O Malley, Glynn Elliott, Jean Marc Zanni, Steven Frick Continos Casting Consortim Members (Ncor, Postech, W Refractories, lgoma, Cors, abein, Mittal Riverdale, aosteel, Steel Dynamics) Matlab Inc. University of Illinois at Urbana-Champaign Metals Processing Simlation ab G Thomas 14