Clear["Global`*"];​
​ω=2000;​
​c1[ζ_]:=-ζω+ω
ζ
2
-1
;​
​c2[ζ_]:=-ζω-ω
ζ
2
-1
;​
​H[ζ_]:=
ω
2
(s-c1[ζ])(s-c2[ζ])
;​
​
In[]:=
StepR[t_,ζ_]:=
ω
2
(c1[ζ]-c2[ζ])
*

c1[ζ]t
-1
c1[ζ]
-

c2[ζ]t
-1
c2[ζ]
(*UnitImpulseIntegratedbyhandUnitstepResponse*)
H1=
ω
2
(s+ω)
2
(*Transferfunctionwhenζ=1*)​
​(*RelevantTimeDomainFunctiont*
-ωt
*u(t)StepR1meanstheIntegralofitasbelow*)
StepR1[t_]:=ω
2
∫
t
0
τ*Exp[-ωτ]*UnitStep[τ]τ​
​
In[]:=
Plot[{StepR[t,0.25],StepR[t,0.5],StepR[t,0.75],StepR1[t],StepR[t,1.5]},{t,0,0.010},PlotRangeAll,
PlotLegends"Expressions"]
In[]:=
0.002
0.004
0.006
0.008
0.010
0.5
1.0
1.5
StepR(t,0.25)
StepR(t,0.5)
StepR(t,0.75)
StepR1(t)
StepR(t,1.5)
Out[]=
Manipulate[Control`PoleZeroPlot[{H[ζ]},PlotLabelStringForm["Pole Zero Plot for ζ = `1`",ζ],PlotLegends
StringForm["ζ = `1` ",ζ],PoleZeroMarkersStyle["x",Large,BackgroundCyan],AxesLabel{"Re","Im"}],{{ζ,
0.5},0,1}]
In[]:=
​
ζ
-1.0
-0.5
0.5
1.0
Re
-1.0
-0.5
0.5
1.0
Im
Pole Zero Plot for ζ = 0.5
ζ = 0.5
Out[]=