Harmonic vibration of an undamped system

by Yi Zhang

This can be found in virtually every dynamics textbook. The undamped system follows the frequency of the excitation, while the displacement response factor R_d.  Response is out of phase when excitation frequency is greater than the natural frequency:  \omega>\omega_0. If there is only one thing to keep in mind of harmonic SDF vibration,  I guess it’s the plot for the response against the frequency ratio. The dynamic object in Mathematica can be built by:

Manipulate[GraphicsRow[
  {GraphicsColumn[{Plot[Rd[r], {r, 0, 3}, AxesOrigin -> {1, 0},
      Frame -> True, PlotRange -> {{0, 3}, {-5, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\!\(\*SubscriptBox[\"R\", \"d\"]\)"},
      Epilog -> {Red, PointSize[Large], Point[{w, Rd[w]}]}],
     Plot[Abs[Rd[r]], {r, 0, 3}, AxesOrigin -> {1, 1}, Frame -> True,
      PlotRange -> {{0, 3}, {0, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "|\!\(\*SubscriptBox[\"R\", \"d\"]\)|"},
      Epilog -> {Red, PointSize[Large], Point[{w, Abs[Rd[w]]}]}]},
    Alignment -> Right, Frame -> All],
   Plot[{1/(1 - w^2) Sin[w t], Sin[w t]}, {t, 0, 4 Pi}, Frame -> True,
     PlotStyle -> {Thick, Dashed},
    PlotRange -> {{0, 4 Pi}, {-10, 10}}, AxesLabel -> {"t", "u"}]
   }, Frame -> All]
 , {{w, 0, "Frequency ratio"}, 0, 3}]

In the right, the dashed curve is the excitation history, while the thick one is for response, which would be out of phase to the excitation when frequency ratio is greater than 1.0 (\omega/\omega_0>1.0).

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