1.37 Plot the dynamic response factor \(R_{d}\) of a system as a function of \(r=\frac {\omega }{\omega _{n}}\) for different damping ratios

Problem: Plot the standard curves showing how the dynamic response \(R_{d}\) changes as \(r=\frac {\omega }{\omega _{n}}\) changes. Do this for different damping ratio \(\xi \). Also plot the phase angle.

These plots are result of analysis of the response of a second order damped system to a harmonic loading. \(\omega \) is the forcing frequency and \(\omega _{n}\) is the natural frequency of the system.

Mathematica

Rd[r_,z_]:=1/Sqrt[(1-r^2)^2+(2 z r)^2]; phase[r_,z_]:=Module[{t}, t=ArcTan[(2z r)/(1-r^2)]; If[t<0,t=t+Pi]; 180/Pi t ]; plotOneZeta[z_,f_] := Module[{r,p1,p2}, p1 = Plot[f[r,z],{r,0,3},PlotRange->All, PlotStyle->Blue]; p2 = Graphics[Text[z,{1.1,1.1f[1.1,z]}]]; Show[{p1,p2}] ]; p1 = Graphics[{Red,Line[{{1,0},{1,6}}]}]; p2 = Map[plotOneZeta[#,Rd]&,Range[.1,1.2,.2]]; Show[p2,p1, FrameLabel->{{"Subscript[R, d]",None}, {"r= \[Omega]/Subscript[\[Omega], n]", "Dynamics Response vs. Frequency\ ratio for different \[Xi]"}}, Frame->True, GridLines->Automatic,GridLinesStyle->Dashed, ImageSize -> 300,AspectRatio -> 1]

p = Map[plotOneZeta[#,phase]&,Range[.1,1.2,.2]]; Show[p,FrameLabel->{{"Phase in degrees",None}, {"r= \[Omega]/Subscript[\[Omega], n]", "Phase vs. Frequency ratio for different \[Xi]"}}, Frame->True, GridLines->Automatic,GridLinesStyle->Dashed, ImageSize->300,AspectRatio->1]