next up previous
Next: 82 Visualize a 2D Up: HOWTO for basic engineering Previous: 80 Solve numerically the

81 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 Matlab 
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]

Image mma_e81_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]

Image mma_e81_2

 
to do


me 2013-01-09