Harmonic vibration of a damped system

Unlike in previous post that phase change could only be or , in the damped system, as the ratio between dynamics response and static response is expressed in the same way as the undamped case

the phase lag varies in . The following Mathematica outputs the displacement response factor, phase lag and normalized time history as the function of damping ratio and frequency ratio .

Rd[w_, \[Zeta]_] := 1/Sqrt[(1 - w^2)^2 + (2 \[Zeta] w)^2]
\[Phi][w_, \[Zeta]_] := -ArcTan[1 - w^2, -2 \[Zeta] w]
ud[w_, \[Zeta]_, t_] :=
 Rd[w, \[Zeta]] Sin[2 Pi (t - \[Phi][w, \[Zeta]]/(2 Pi))]
us[t_] := Sin[2 Pi t]
Manipulate[
 GraphicsRow[{GraphicsColumn[{Plot[Rd[w, \[Zeta]], {w, 0, 3},
      AxesOrigin -> {1, 1}, PlotRange -> {{0, 3}, {0, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\!\(\*SubscriptBox[\"R\", \"d\"]\)"},
      Epilog -> {PointSize[Large], Red, Point[{w, Rd[w, \[Zeta]]}]},
      Ticks -> {{0, 1, 2, 3}, {0, 1, 2, 3, 4, 5}}],
     Plot[\[Phi][w, \[Zeta]], {w, 0, 3}, AxesOrigin -> {1, Pi/2},
      PlotRange -> {{0, 3}, {0, Pi}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\[Phi]"},
      Epilog -> {PointSize[Large], Red,
        Point[{w, \[Phi][w, \[Zeta]]}]},
      Ticks -> {{0, 1, 2, 3}, {0, Pi/2, Pi}}]}, Frame -> All],
   Plot[{ud[w, \[Zeta], t], us[t]}, {t, 0, 2},
    PlotStyle -> {Thick, Dashed},
    AxesLabel -> {"t/T",
      "\!\(\*SubscriptBox[\"u\", \"d\"]\)/\!\(\*SubscriptBox[\"u\", \
\"st\"]\)"}]}, Frame -> All], {{w, 0.5, "Frequency ratio"}, 0,
  3}, {{\[Zeta], 0.2, "Damping ratio"}, 0, 1}]

Special cases include

1. Long period excitation, i.e., , gives a pseudo-static response. In this case, the system “waits” until it “feels” the excitation completely. is greater but very close to , and the displacement is essentially in phase with excitation force, in other words, dynamic effect is near to none.
   

   Long period/low frequency loading response

2. Short period excitation gives very small , though the phase lag is . Here the system barely reacts when the load is reversed, thus leads to small displacement.
   

   Short period/high frequency loading response

3. Resonant period, i.e., , leads to . Now is very sensitive to damping change, namely, the response is controlled by the damping: a small change of damping ratio leads to great reaction of the structure. When , we have .
   

   Resonant frequency: small damping

   

   Resonant frequency: medium damping

   

   Resonant frequency: large damping

The last case is the what’s essentially behind the viscous damping devices applied to buildings like this.

Harmonic vibration of an undamped system

This can be found in virtually every dynamics textbook. The undamped system follows the frequency of the excitation, while the displacement response factor .  Response is out of phase when excitation frequency is greater than the natural frequency:  . 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 ().