Mesh Mess

FEM, CFD, CAE, and all that.

Tag: earthquake

Harmonic vibration of a damped system

Unlike in previous post that phase change could only be 0 or \pi, in the damped system, as the ratio between dynamics response and static response is expressed in the same way as the undamped case
\frac{u_d}{u_{s0}}=R_d\sin(\omega t-\phi),
the phase lag \phi varies in [0,\pi]. The following Mathematica outputs the displacement response factor, phase lag and normalized time history as the function of damping ratio \zeta and frequency ratio w=\omega/\omega_0.

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]
 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., \omega\ll\omega_0, gives a pseudo-static response. In this case, the system “waits” until it “feels” the excitation completely. R_d is greater but very close to 1, 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 R_d, though the phase lag is \pi. 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., \omega\approx\omega_0, leads to \phi\approx\pi/2. Now R_d is very sensitive to damping change, namely, the response is controlled by the damping: a small change of damping ratio \zeta leads to great reaction of the structure. When \omega=\omega_0, we have R_d=1/(2 \zeta) .

    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 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:

  {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).

BBC science news on Christchurch quake

Here is a very interesting report on Christchurch quake happened last month, addressing the two major causes that have contributed to the damage in the city. The technique used in the mapping of the earth movement, called “synthetic aperture radar inteferogram”, gives a very precise (cm scale)  movements mapping of the landscape around the city. The damage was largely due to closeness of epicenter, located at the southeast suburb.  Another cause was the geostructure, in particular the liquifaction, the effect of which is demonstrated in the picture below (from the same report).

Preparing for the Big One:Audio from the National Building Museum

Preparing for the Big One: Assessing American Building Codes

“The 2010 earthquakes in Haiti and Chile reinforced the importance of building codes. Are regions in the United States just as vulnerable to a catastrophic earthquake? David Applegate, Senior Science Advisor for Earthquake and Geologic Hazards at the U.S. Geological Survey; Michael J. Armstrong, Senior Vice President, International Code Council; and Michael Mahoney, Geophysicist, Federal Emergency Management Agency discuss the state of seismic building codes around the country. Tom Ichniowski, DC Bureau Chief for Engineering News-Record, moderates.”

“Bad buildings kill them”

A troubling fact is that some HARD evidences are required before people learn a lesson which is learned before by others, at other location, even in a recent past. It is less than two years after China’s quake in 2008 taking over 80,000 lives, and now Haiti. I am reading almost the exact comments about Haiti earthquake as I did two years ago about China. What is in common for the two cases? They are at the poorest region. Unlike Haiti, China may be at a better position of infrastructure construction and building code implementation. However, the region being attacked two years ago is one of most underdeveloped regions in southwest China, where the building codes were hardly enforced.

It is the human nature (or government?) of ignoring risks that are not imminent, even when we are presented with dead cold records.