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# Physical System Modelling, Time Response And Stability (Math Problem Sample)

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THE TASK WAS TO SOLVE CONTROL SYSTEMS MATH PROBLEMS AND SIMULATE USING MATLAB SOFTWARE

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PHYSICAL SYSTEM MODELING, TIME RESPONSE AND STABILITY
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Physical System Modeling, Time Response and Stability
1 A differential equation that relates ea(t) and θL(t)
We know that eat=Raia(t)+Kbdθm(t)dt and θmt=N2N1θL(t)
dθm(t)dt =ddt(θmt)=N2N1dθL(t)dt
Substituting the value of dθm(t)dt in eat gives:
eat=KbN2N1dθLtdt+Raia(t)
Therefore, eat=Raia(t)+(KbN2N1dθL(t)dt)
2 GS=θL(S)Ea(S) of the system
eat=Raia(t)+Kbdθm(t)dt in the s-Domain becomes Eas=RaIa(s)+KbSθm(s)
Also, TmS=KtIa(s), IaS=TmSKt
EaS=RaTmSKt+Kbsθms………..(i)
We also know that TmS=(Jms2+Dms)θms………..(ii)
Substituting equation (ii) in equation (i)
EaS=RaKt(Jms2+Dms)θms+Kbsθms
EaS=sθmsRaKt(Jms+Dm+Kb)
After simplification, θmsEaS is found to be:
θmsEaS=Kt(RaJm)s+1Jm(Dm+KtK

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