### Harmonic vibration of a damped system

#### by Yi Zhang

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

- 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.
- 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.
- 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 .

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

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